The Bulletin of the International Linear Algebra Society IMAGE · The Bulletin of the International Linear Algebra Society IMAGE Serving the International Linear Algebra Community

The Bulletin of the International Linear Algebra Society

IMAGEServing the International Linear Algebra Community

Issue Number 43, pp. 1-44, Fall 2009

Editor-in-Chief: Jane M. Day, Dept. of Mathematics, San Jose State University, San Jose, CA, USA 95192-0103; [email protected] Editors: Oskar Maria Baksalary, Steven J. Leon, Peter Šemrl, James Weaver and Fuzhen Zhang

Past Editors-in-Chief: Robert C. Thompson (1988); Jane M. Day & R.C. Thompson (1989);Steven J. Leon & R.C. Thompson (1989-1993); Steven J. Leon (1993-1994); Steven J. Leon & George P.H. Styan (1994-1997);

George P.H. Styan (1997-2000); George P.H. Styan & Hans J. Werner (2000-2003); Bryan Shader & Hans J. Werner (2003-2006); Jane M. Day and Hans J. Werner (2006-2008)

Upcoming Conferences and WorkshopsOther Conferences of Interest……………………………………………………………………………………………………………..2 16th ILAS Conference, Pisa, Italy, June 21-25, 2010………………………………………………………………………….…………...3Applied Linear Algebra Conference in Honor of Hans Schneider, Novi Sad, Serbia, May 24-28, 2010………………………………....519th International Workshop on Matrices and Statistics, Shanghai, China, June 5-8, 2010……………..…….…………………………..5International Summer School on Numerical Linear Algebra, Fasano, Italy, June 7-18, 2010…………………………………………….64th Annual International Conference on Mathematics and Statistics, Athens, Greece, June 14-17, 2010………………………………....710th Workshop on Numerical Range and Numerical Radii, Krakow, Poland, June 27-29, 2010……………………………………….....7Zero-One Matrix Theory and Related Topics, Coimbra, Portugal, June 17-19, 2010………………………………………………….….7International Workshop on Accurate Solution of Eigenvalue Problems VIII, Berlin, Germany, June 28-July 1, 2010……………….......819th International Symposium on Mathematical Theory of Networks and Systems, Budapest, Hungary, July 5-9, 2010…………….......89th China Matrix Theory and Applications International Conference, Shanghai, China, July 18-22, 2010……………………………….9International Conference on Trends and Perspectives in Linear Statistical Inference, Tomar, Portugal, July 27-31, 2010……………,....9International Conference on Recent Trends in Graph Theory and Combinatorics, Cochin, India, August 12-15, 20………….........…..102nd IMA Conference on Numerical Linear Algebra and Optimization, Birmingham, UK, September 13-15, 20…………………….......10

Reports on Conferences and Workshops Workshop on Spectral Graph Theory with Applications, Rio de Janeiro, Brazil, December 1-4, 2008………………………………..,...1215th Haifa Matrix Theory Conference, Haifa, Israel, May 18-21, 2009………………………………………………………………,….13Summer School and Advanced Workshop on Trends and Developments in Linear Algebra, Trieste, Italy, June 22-July 3, 2009……...14International Workshop on Scientific Computing and Its Applications, Shanghai, China, June 28-30, 2009…………………………...1510th SIAM Applied Linear Algebra Conference, Monterey, CA, USA, October 26-29, 2009……………………………...…………….17

ObituaryIsrael Gohberg, 1928-2009, by Leiba Rodman………………………………………………………………………………..……….…18

Articles Addendum to “Linear Algebra in Iran” by Mehdi Radjabalipour and Heydar Radjavi, IMAGE Issue 42, 2009…………………….….19Finding John Francis Who Found QR Fifty Years Ago, by Frank Uhlig………………………………………………………………....19Symmetric Sudoku Matrices and Magic Squares, by Richard William Farebrother………………………………….………………….23Ancient Approximations To Pi, by Richard William Farebrother…………………………………………………………….………......24The GAMM Activity Group Applied and Numerical Linear Algebra, by Peter Benner and Daniel Kressner…………………..………..25.Book ReviewMatrix Mathematics: Theory, Facts and Formulas, by Dennis S. Bernstein, 2nd ed., 2009; reviewed by Helmut Lütkepohl…………....28

Quick Notes about BooksNew Books……………………………………………………………………….………………………………………………………29List of Books on the ILAS-IIC Website, O. Baksalary………………………………………………………………………………..…29

Contents continued on page 2

Page 2 Fall 2009: IMAGE 43

Other Conferences of Interest

January 13-16, 2010, AMS/MAA Joint Meetings, San Francisco, CA, USA. Includes special sessions on Inverse Problems, Functional Analysis and Operator Theory, and Innovative and Effective Ways to Teach Linear Algebra

July 12-16, 2010, SIAM Annual Meeting, Pittsburgh, PA, USA. Includes a Linear Algebra track; deadline for minisymposia proposals is January 12, 2010

August 19-27, 2010, International Congress of Mathematicians, ICM-2010, Hyderabad, India

July 18-22, 2011, International Conference on Industrial and Applied Mathematics, Vancouver, British Columbia, Canada

August 22 -26, 2011, 17th ILAS Conference, Braunschweig, Germany

June 18-22, 2012, SIAM Conference on Applied Linear Algebra, Valencia, Spain

Journal NewsCall for Papers: Special Issue of LAA on Tensors and Multilinear Algebra………………………………………………………...29Call for Papers: Special Issue of LAA on 1st Montreal Workshop on Idempotent and Tropical Mathematics……………………...30Call for Papers: JP Journal of Algebra, Number Theory and Applications………….…………………………………..………......30 Call for Papers: Journal of Algebra, Number Theory: Advances and Applications………………………………………………....30

ILAS News ILAS President/Vice President Annual Report……………………………………………………………………………………...30Increase in ILAS Dues………………………………………………………………………………………………………………32Nominations for 2009 ILAS Elections……………………………………………………………………………………………....32New Address for ILAS President Steve Kirkland…………………………………………………………………………………...32ILAS Members Named SIAM Fellows……………………………………………………………………………………………...33Call for News for IMAGE Issue 44……………..…………………………………………………………………………………...33

EmploymentResearch Positions at SISTA, University of Leuwen, Belgium………………………………..…………………………………....33

IMAGE Problem Corner: Solutions of Old ProblemsProblem 41-4: Integer Matrices with Unit Determinants II………………………………………………………………………....35Problem 42-1: Square Root of a Product of Two Orthogonal Projectors………………………………………………………..…..37Problem 42-2: Properties of a Certain Matrix Product…………………………………………………………………………........37Problem 42-3: Commutators and Nonderogatory Matrices……………………………………………………………………..…..38Problem 42-4: Eigenvalue Interlacing for Normal Matrices………………………………………………………………………...41Problem 42-5: Obtuse Basis and Acute Dual Basis for Euclidean Space……………………………………………………….......42Problem 42-6: Positive Semidefinite Matrices……………………………………………………………........................................43

IMAGE Problem Corner: New ProblemsProblem 43-1: Characterizations of EP Matrices………………………………………………........................................................44Problem 43-2: Square Roots of (Skew)-Involutory Matrices……………………………………………………………………….44Problem 43-3: Reconstructing a Positive Semidefinite Matrix……………………………………………………………………...44Problem 43-4: A Trace Inequality for Positive Semidefinite Matrices……………………………………………………………....44Problem 43-5: The Matrix Equation YTY = Y…………………………………………………………………………………..…...44Problem 43-6: Matrix Similarity………………………………………………………………………………………………….....44Problem 43-7: Three Positive Semidefinite Matrices………………………………………………………………………………..44


UPCOMING CONFERENCES AND WORKSHOPS

The 16th ILAS ConferencePisa, Italy

June 21-25, 2010

The 16th ILAS Conference will be held in Pisa, Italy. It will open at 9:00 on Monday June 21 and close at 18:00 on Friday June 25. The Plenary Speakers will be Rajendra Bhatia, Richard A. Brualdi, Pauline van den Driessche, Nick Higham, Beatrice Meini, Vadim Olshevsky, Zdenek Strakos, Daniel B. Szyld, and Luis Verde Star.

Special Lectures will be given by Oliver Ernst (SIAG/LA), Olga Holtz (Taussky-Todd), Lek-Heng Lim (LAA), Cleve Moler (Hans Schneider Prize), and Beresford N. Parlett (Hans Schneider Prize).

Six minisymposia will highlight the recent developments in specific areas of linear algebra and its applications. The titles and organizers are:

§Structured Matrices (Yuli Eidelman, Lothar Reichel, Marc Van Barel)§Markov Chains (Steve Kirkland, Michael Neumann)§Nonnegative Matrices (Judi McDonald, Michael Tsatsomeros)§Matrix Functions and Matrix Equations (Chun-Hua Guo, Valeria Simoncini)§Combinatorial Linear Algebra (Shaun Fallat, Bryan Shader)§Linear Algebra Education (Avi Berman, Steven J. Leon)§Matrix Inequalities (Chi-Kwong Li and Fuzhen Zhang)

The Scientific Committee consists of Avi Berman, Michele Benzi, Dario A. Bini, Luca Gemignani, Leslie Hogben, Steve Kirkland, Julio Moro, Ilia Spitkovsky, Francoise Tisseur, and Eugene Tyrtyshnikov. The Local Organizing Committee is Dario A. Bini (Chair), Gianna Del Corso, Bruno Iannazzo, Beatrice Meini, Ornella Menchi, and Federico Poloni.

The conference proceedings will appear as a special issue of Linear Algebra and its Applications. The Guest Editors for this issue are: D. A. Bini, A. Boettcher, L. Gemignani, L. Hogben, and F. Tisseur.

The sponsors of this conference are the Universita' di Pisa Dipartimento di Matematica "Leonida Tonelli", Dipartimento di Matematica Applicata "Ulisse Dini", Dipartimento di Informatica, Centro Interdipartimentale "E. Piaggio", and Facolta' di Economia; Taylor & Francis Group; ILAS; and the SIAM Activity Group on Linear Algebra.

The deadline for early registration is January 31, 2010. The registration fee is 220 Euros if paid by January 31 and 260 Euros otherwise. For students, the registration fee is 160 Euros if paid by January 31 and 260 Euros otherwise. For details about registration, accommodations, program, etc., visit the conference website http://www.dm.unipi.it/~ilas2010/index.php?page=proceed. For more information, send email to [email protected] with subject ilas2010.

Journal NewsCall for Papers: Special Issue of LAA on Tensors and Multilinear Algebra………………………………………………………...29Call for Papers: Special Issue of LAA on 1st Montreal Workshop on Idempotent and Tropical Mathematics……………………...30Call for Papers: JP Journal of Algebra, Number Theory and Applications………….…………………………………..………......30 Call for Papers: Journal of Algebra, Number Theory: Advances and Applications………………………………………………....30

ILAS News ILAS President/Vice President Annual Report……………………………………………………………………………………...30Increase in ILAS Dues………………………………………………………………………………………………………………32Nominations for 2009 ILAS Elections……………………………………………………………………………………………....32New Address for ILAS President Steve Kirkland…………………………………………………………………………………...32ILAS Members Named SIAM Fellows……………………………………………………………………………………………...33Call for News for IMAGE Issue 44……………..…………………………………………………………………………………...33

EmploymentResearch Positions at SISTA, University of Leuwen, Belgium………………………………..…………………………………....33

IMAGE Problem Corner: Solutions of Old ProblemsProblem 41-4: Integer Matrices with Unit Determinants II………………………………………………………………………....35Problem 42-1: Square Root of a Product of Two Orthogonal Projectors………………………………………………………..…..37Problem 42-2: Properties of a Certain Matrix Product…………………………………………………………………………........37Problem 42-3: Commutators and Nonderogatory Matrices……………………………………………………………………..…..38Problem 42-4: Eigenvalue Interlacing for Normal Matrices………………………………………………………………………...41Problem 42-5: Obtuse Basis and Acute Dual Basis for Euclidean Space……………………………………………………….......42Problem 42-6: Positive Semidefinite Matrices……………………………………………………………........................................43

IMAGE Problem Corner: New ProblemsProblem 43-1: Characterizations of EP Matrices………………………………………………........................................................44Problem 43-2: Square Roots of (Skew)-Involutory Matrices……………………………………………………………………….44Problem 43-3: Reconstructing a Positive Semidefinite Matrix……………………………………………………………………...44Problem 43-4: A Trace Inequality for Positive Semidefinite Matrices……………………………………………………………....44Problem 43-5: The Matrix Equation YTY = Y…………………………………………………………………………………..…...44Problem 43-6: Matrix Similarity………………………………………………………………………………………………….....44Problem 43-7: Three Positive Semidefinite Matrices………………………………………………………………………………..44

The Johns hopkins UniversiTy press1-800-537-5487 • www.press.jhu.edu

Linear aLgebraChallenging Problems for Students

second editionFuzhen Zhang

Linear algebra is a prerequisite for students majoring in mathematics and is required of many undergraduate and first-year graduate students in statistics, engineering, and related areas. This fully updated and re-vised text defines the discipline’s main terms, explains its key theorems, and provides over 425 example problems ranging from the elementary to some that may baffle even the most seasoned mathematicians. Vital concepts are highlighted at the beginning of each chapter and a final section contains hints for solving the problems as well as solutions to each example.

Based on Fuzhen Zhang’s experience teaching and researching algebra over the past two decades, Linear Algebra is the perfect examination study tool. Students in beginning and seminar-type advanced linear algebra classes and those seeking to brush up on the topic will find Zhang’s plain discussions of the subject’s theories refreshing and the problems diverse, interesting, and challenging.

$25.00 paperback

“Mathematics is not a spectator sport. Actively working on well-chosen problems is the best way to understand and master any area of mathematics. Students at all levels will benefit from putting pencil to paper and solving Professor Zhang’s linear algebra problems.” —Roger Horn, University of Utah

“This is a large collection of good problems of varied levels of difficulty. They offer readers many opportunities for thinking of clever ways to apply classic theorems.” —Jane Day, San Jose State University


Applied Linear Algebra Conference in Honor of Hans SchneiderDepartment of Mathematics, Faculty of Science, Novi Sad, Serbia

May 24-28, 2010

This conference on Applied Linear Algebra, in honor of Hans Schneider, is inspired by the success of two previous conferences, Applied Linear Algebra in honor of Richard Varga, 2005, Palic; and Applied Linear Algebra in honor of Ivo Marek, 2008, Novi Sad. ALA 2010 has a similar purpose, to review the numerous contributions of the honree and to report and discuss recent progress in the field through the participation of international leaders who will gather in his honor. In addition, the 10th GAMM Workshop on Applied and Numerical Linear Algebra with special emphasis on Positivity will be organized as a part of ALA 2010. A special issue of Linear Algebra and its Applications will be devoted to selected papers presented during the conference.

For more information, visit http://www.dmi.uns.ac.rs/events/ala2010.

The 19th International Workshop on Matrices and Statistics will be held June 5-8, 2010, at Shanghai Finance University, Shanghai. The purpose is to stimulate research and, in an informal setting, to foster the interaction of researchers in the interface between statistics and matrix theory. The workshop will provide a forum through which statisticians and mathematicians may be better informed of the latest theoretical developments and techniques as well as their use in related application areas. Participants will be able to take advantage of the opportunity to exchange ideas with researchers from a number of countries, who are working in these areas.

The conference seeks a wide variety of invited and contributed papers. Special sessions on computational matrix procedures, statistical applications in economics and matrix methods in applied probability are proposed. A list of invited speakers and dates for submission of abstracts and registration will be available shortly on the workshop website: http://www1.shfc.edu.cn/iwms/index.asp. Submission of abstracts and registration will also be done through the website. Publication of the workshop proceedings is being explored. For enquiries regarding local arrangements, use the email address: [email protected]

Social activities, including an excursion and workshop banquet, will be held. In addition, a pre-workshop 14-day

Tour of China, starting in Beijing on Friday May 21, has been arranged. Details may be found on the workshop website.

The International Organizing Committee (IOC) is George P. H. Styan, Honorary Chair of IOC and Honorary Chair of ISC (Canada)*; Zhong-Ci Shi, Honorary Chair of IOC, Chair of ISC (China)*; Jeffrey J. Hunter, Chair of the IOC (New Zealand)*; Hans Joachim Werner, Vice-Chair of the IOC (Germany)*; S. Ejaz Ahmed (Canada); Zhi-Dong Bai (Singapore)*; Zhong-Zhi Bai (China)*; Guoliang Chen (China)*; Ben-Yu Guo (China)*; Er-Xiong Jiang (China)*; Tian-Gang Lei (China)*; Augustyn Markiewicz (Poland); Simo Puntanen (Finland); Götz Trenkler (Germany); Julia Volaufova (USA); Dietrich von Rosen (Sweden); Musheng Wei (China)*; and Xue-Jun Xu (China)*. *Denotes those who are also members of the International Scientific Committee (ISC).

The Local Organizing Committee is Yonghui Liu (China); Chair, Rong-Xian Yue, Co-Chair (China); Yongge Tian; Baoxue Zhang; Rui Li; Rongqiang Che; Yong Fang; Manrong Wang; Qian Wang; and Lei Fu.

Sponsors of the workshop include Shanghai Finance University, National Natural Science Foundation of China, Shanghai Normal University, and the Division of Scientific Computation E-Institute of Shanghai Universities. The International Linear Algebra Society endorses IWMS-2010.

19th International Workshop on Matrices and StatisticsShanghai Finance University, Shanghai, China

June 5-8, 2010


SIAM International Summer School on Numerical Linear Algebra (ISSNLA)Fasano, Italy

June 7-18, 2010

This Summer School, co-sponsored by SIAM and the Instituto per le Applicazioni del Calcolo, will take place at Selva di Fasano, Brindisi, Italy, during June 7-18, 2010. Fasano is located in southern Italy, very near the Adriatic Sea.

This is the second ISSNLA, which coincides with the first Gene Golub Summer School, funded by Gene Golub’s legacy to SIAM. These ISSNLA workshops are organized by the SIAM Activity Group on Linear Algebra. The main goal is to offer accessible courses on current developments in Numerical Linear Algebra and its applications to other disciplines.

The following four courses will be given in 2010:

• Minimizing communication in numerical linear algebra, James Demmel, University of California at Berkeley, USA

• Nonlinear eigenvalue problems: analysis and numerical solution, Volker Mehrmann, Technische Universitaet Berlin, Germany

• From Matrix to Tensor: The Transition to Computational Multilinear Algebra, Charles Van Loan, Cornell University, Ithaca, New York, USA

• Linear Algebra and Optimization, Margaret H. Wright, Courant Institute, New York University, USA

These courses will focus on recent research topics that have reached a significant maturity and whose impact is widely recognized by the community, but that are not usually included in text books or in basic courses at the doctoral level. The workshop is mainly intended for doctoral students in any field where methods and algorithms of Numerical Linear Algebra are used.

The summer school is geared towards graduate students, with a limit of 50 students. There will be no registration fee. Funding for local accommodations and/or local expenses will be available for some of the participants. Limited travel funds may also be available.

The deadline for applications is February 1, 2010. For more details, watch the conference website http://www.ba.cnr.it/ISSNLA2010.

The Advisory Committee consists of: Dario Bini (Università di Pisa, Italy) Jim Demmel (University of California at Berkeley, USA)Iain Duff (Rutherford Appleton Laboratory, UK)Anne Greenbaum (University of Washington, USA)Martin Gutknecht (ETH Zurich, Switzerland)Nick Higham (University of Manchester, UK)Rich LeHoucq (Sandia National Lab, Albuquerque, USA)John Lewis (Cray Inc., USA)Volker Mehrmann (Technische Universität Berlin, Germany)Dianne O’Leary (University of Maryland, USA)Michael Overton (New York University, USA)Lothar Reichel (Kent State University, USA)Axel Ruhe (KTH Stockholm, Sweden)Henk van der Vorst (Utrecht University, The Netherlands)Paul van Dooren (Catholic University of Louvain, Belgium).

The Steering Committee consists of Froilán Dopico (Universidad Carlos III de Madrid, Spain), Andreas Frommer (Bergische Universität Wuppertal, Germany), Ilse Ipsen (North Carolina State University), and Daniel Szyld (Temple University). The Local Organizing Committee members are Fasma Diele (IAC-CNR), Teresa Laudadio (IAC-CNR), Nicola Mastronardi (IAC-CNR), Filippo Notarnicola (IAC-CNR), Valeria Simoncini (Università di Bologna), and Nicola Fracasso (Administrator/Manager IAC-CNR).


University of Coimbra Library

Zero-One Matrix Theory and Related TopicsCoimbra, PortugalJune 17-19, 2010

This meeting will be held at the University of Coimbra. The invited speakers will be Alexander Barvinok (University of Michigan, USA), Adrian Bondy (Université Claude Bernard Lyon 1, France), Richard A. Brualdi (University of Wisconsin-Madison, USA), Domingos M. Cardoso (Universidade de Aveiro, Portugal), Geir Dahl (University of Olso, Norway), Michael Drmota (Technische Universität Wien, Austria), Catherine Greenhill (University of New South Wales, Australia), Gyula O. H. Katona (Alfréd Rényi Institute of Mathematics, Hungary), Hadi Kharaghani (University of Lethbridge, Canada), Christian Krattenthaler (Universität Wien, Austria), Brendan McKay (Australian National University, Australia), Irene Sciriha (University of Malta, Malta), and Herbert S. Wilf (University of Pennsylvania, USA) - ILAS Lecturer.

Contributed papers are still sought. For more information about the conference, see Image issue 42, p. 8, and visit the conference website http://www.mat.uc.pt/~cmf/01MatrixTheory.

This meeting is endorsed by the International Linear Algebra Society and sponsored by the Center for Mathematics and the Department of Mathematics of the University of Coimbra, and by the American Mathematical Society.

4th Annual International Conference on Mathematics & Statistics

Athens, GreeceJune 14-17, 2010

The Mathematics and Statistics Research Unit of the Athens Institute for Education and Research (ATINER) will hold its 4th International Conference in Athens, Greece, June 14-17, 2010. Papers from all areas of Mathematics, Statistics, Mathematics and Engineering and Mathematics and Education are welcome.

Selected reviewed papers will be published in a special volume of the Conference Proceedings. To contribute, send an abstract of about 300 words by December 14, 2009, to Dr. Vladimir Akis ([email protected]). For more information, see the conference website www.atiner.gr/docs/Mathematics.htm.

10th Workshop on Numerical Ranges and Numerical RadiiKrakow, PolandJune 27-29, 2010

The 10th Workshop on “Numerical Ranges and Numerical Radii” will be held in the beautiful old city Krakow, Poland from June 27-29, 2010, immediately following the 2010 ILAS Conference at Pisa, Italy, June 21-25, 2010. The workshop is endorsed by ILAS.

The purpose of the workshop is to stimulate research and foster interaction of researchers interested in the subject. The informal workshop atmosphere will facilitate the exchange of ideas from different research areas and, hopefully, the participants will leave informed of the latest developments and newest ideas.

The Organizers are Chi-Kwong Li, College of William and Mary; Franciszek Hugon Szafraniec, Uniwersytet Jagiellonski; and Jaroslav Zemanek, Instytut Matematyczny PAN. There will be no registration fee. A special issue of Linear and Multilinear Algebra will be devoted to the papers presented at the workshop. For more information, visit the conference website, http://www.math.wm.edu/~ckli/wonra10.html. To see some background about the subject and previous meetings, visit the website http://www.math.wm.edu/~ckli/wonra.html .


19th International Symposium onMathematical Theory of Networks and Systems

Budapest, HungaryJuly 5-9, 2010

The 19th International Symposium on Mathematical Theory of Networks and Systems (MTNS 2010) will take place at ELTE, the University Congress Center, in Budapest, Hungary, July 5-9, 2010. The Symposium is hosted by the Eötvös Loránd University (ELTE) and the Computer and Automation Research Institute of the Hungarian Academy of Sciences (MTA SZTAKI).

MTNS 2010 is a prime conference in the general area of mathematical system theory. The symposium is interdisciplinary and attracts mathematicians, engineers and researchers working in any aspect of system theory and its applications. A prime objective of the MTNS 2010 Symposium is to explore and present mathematics as a key technology for the 21st century.

The scientific programme will cover 12 main areas: Biological Systems, Communication Systems, Computing, Control and Systems Theory, Cooperative Systems, Economics and Systems Theory, Hybrid Systems, Mechanical Systems, Networked Control, Signal

Processing, Stochastic Systems, and Systems Inspired Mathematics.

For further information, please visit the conference web site: www.conferences.hu/mtns2010

The General Chair of the conference is Prof. György Michaletzky and the IPC Chair is Prof. László Gerencsér.

International Workshop on Accurate Solution of Eigenvalue Problems VIIIBerlin, Germany

June 28-July 1, 2010

IWASEP VIII will be held June 28-July 1, 2010, in cooperation with the GAMM Activity Group Applied and Numerical Linear Algebra, at TU Berlin. Its purpose is to bring together experts on accuracy issues in the numerical solution of eigenvalue problems for research presentations and discussions. There will be no parallel sessions; research presentations will be 30 to 40 minutes each, with sufficient time for discussions. Specially titled discussion/working groups and a poster session will also be organized.

The invited speakers are Rafikul Alam, Indian Inst. of Technology Guwahati; Peter Benner, Chemnitz University of Technology; James Demmel, University of California at Berkeley; Louis Komzsik, Siemens PLM Software; Wen-Wei Lin, National Chiao Tung University (TBC); Julio Moro, Universidad Carlos III de Madrid; Beresford Parlett, University of California at Berkeley; Valeria Simoncini, University of Bologna; Pete Stewart, University of Maryland; Eugen Tyrtyshnikov, Russian Academy of Sciences; Nick Trefethen, Oxford University; and Marc Van Barel, Katholieke Universiteit Leuven.

The organizers are Jesse Barlow, Penn. State Univ., USA; Zlatko Drmac, Univ. of Zagreb, Croatia; Volker Mehrmann, TU Berlin, Germany, (Chair); Ivan Slapnicar, Univ. of Split, Croatia; and Kresimir Veselic, Fernuniversitaet, Hagen, Germany.

Linear Algebra and its Applications has published three special issues related to the previous workshops in this series: 309/1-3 (2000), 358 (2003), and part of 417/2-3 (2006). Also the SIAM Journal on Matrix Analysis and Applications published a special issue in Vol. 28/4 (2006), and reports on those workshops were published in SIAM News.

The deadline for abstracts and for registration is April 1, 2010. The organizers welcome proposals for talks and posters that are consistent with the theme of the meeting. A prize will be given for the best poster. For more information, visit http://www3.math.tu-berlin.de/iwasep8/.

View of Budapest and Chain Bridge across the Danube


The International Conference on Trends and Perspectives in Linear Statistical Inference

LinStat2010Tomar, PortugalJuly 27-31, 2010

LinStat 2010 will be held in Tomar, Portugal. The purpose of this conference is to bring together researchers to discuss current developments in a variety of aspects of statistics and its applications, and also to encourage young scientists. Prizes will be awarded to graduate students or scientists with a recently completed Ph.D., for the best poster and the best talk.

Young Scientists awarded at Linstat 2008, which was held in Będlewo, Poland, will be Invited Speakers at LinStat 2010.

The conference organizers are Joao T. Mexia (Scientific Committee) and Francisco Carvalho (Organizing Committee).

To view the invited speakers and other information, visit the conference website http://www.linstat2010.ipt.pt.

19th China Matrix Theory and Applications International Conference

Shanghai University, Shanghai, ChinaJuly 18 - 22, 2010

The purpose of this conference is to stimulate research and to foster the interaction of researchers. The topics cover matrix theory, linear algebra and applications, including traditional linear algebra, combinational linear algebra and numerical linear algebra. The conference will provide a forum through which scientists may be better informed of the latest developments and newest techniques and may exchange ideas with researchers from several different countries.

The invited speakers are: Man-Duen Choi, University of Toronto, Toronto, Canada; Yongdo Lim, Kyungpook National University, Taegu, Republic of Korea; Koenraad M.R. Audenaert, University of London, Egham, United Kingdom; Chi Kwong Li, Ferguson Professor of Mathematics, College of William & Mary, Williamsburg, VA, USA; PeiYuan Wu, National Chiao Tung University, Hsinchu, Taiwan; Zhaojun Bai, University of California at Davis, Davis, CA USA; Qingwen Wang, Shanghai University, Shanghai, China; Ming Gu, University of California at Berkeley, CA USA; Daniel Kressner, ETH Zurich, Switzerland; Biswa Nath Datta, Northern Illinois University, De Kalb, IL USA; Natalia Bebiano, University of Coimbra, Coimbra, Portugal; and Zhinan Zhang, Xinjiang University, China.

The members of the Organizing Committee are: Shufang Xu, Peking University, Beijing, China; Yimin Wei, Fudan University, Shanghai , China; Xingzhi Zhan, East China Normal University, Shanghai, China; Yongge Tian, China Central University of Finance and Economy, Beijing, China; Hongguo Xu, University of Kansas, Lawrence, KS USA; Rencang Li,University of Texas, Arlington, TX USA; Erxiong Jiang (Chair), Shanghai University, Shanghai, China; Chuanqing Gu, Shanghai University, Shanghai, China; Zhongzhi Bai, Chinese Academy of Science, Beijing, China.

The members of Local Organizing Committee are: Qingwen Wang, (Chair) Shanghai University, Shanghai, China; Xinjian Xu, Shanghai University, Shanghai, China; Qin Zhang, Shanghai University, Shanghai, China; Xiaomei Jia, Shanghai University, Shanghai, China; Jianxin Yang, Shanghai University, Shanghai, China; Jianjun Zhang, Shanghai University, Shanghai, China; Xiaodong Zhang, Jiaotong University, Shanghai, China; Yonghui Liu, Shanghai Finance University, Shanghai, China; Changqing Xu, Zhejiang Forestry University, Hangzhou, China.

For details, visit the conference web page http://math.shu.edu.cn/CLAS2010/Default.aspx.

Participants wishing to present a contributed talk should submit by e-mail an extended abstract (1-2 pages) written in Latex to Prof. Qin Zhang ([email protected] or [email protected]). The deadline for submissions is April 30, 2010. Notification of acceptance will be given by June 10, 2010.

Shanghai University Main Gate


Mural Mattancherry Palace,

Cochin, India

International Conference on Recent Trends in Graph Theory and Combinatorics

(ICRTGC-2010)Cochin, India

August 12-15, 2010

The Conference on Recent Trends in Graph Theory and Combinatorics will be held August 12-15, 2010, in Cochin, India. This is a Satellite Conference of the International Congress of Mathematicians, ICM-2010, which will be held at Hyderabad, India on August 19-27, 2010.

The Covener of ICRTGC-2010 is Ambat Vijayakumar, Department of Mathematics, Cochin University of Science and Technology, Cochin-682 022, India. Send email to [email protected] or [email protected]. For more information, see Image issue 42, p. 9 and visit the conference website http://ams.org/mathcal/info/2010_aug12-15_cochin.html.

2nd IMA Conference on Numerical Linear Algebra and Optimization

University of Birmingham, UKSeptember 13-15, 2010

The Institute of Mathematics and its Applications (IMA) and the University of Birmingham are pleased to announce the Second IMA Conference on Numerical Linear Algebra and Optimization, Sept. 13-15, 2010. Future meetings will be held biennially. The meeting is co-sponsored by SIAM and ILAS, whose members will receive the IMA members’ registration rate.

The success of modern codes for large-scale optimization is heavily dependent on the use of effective tools of numerical linear algebra. On the other hand, many problems in numerical linear algebra lead to linear, nonlinear or semidefinite optimization problems. The purpose of the conference is to bring together researchers from both communities and to find and communicate points and topics of common interest.

Conference topics include any subject that could be of interest to both communities, such as direct and iterative methods for large sparse linear systems, eigenvalue computation and optimization, large-scale nonlinear and semidefinite programming, effect of round-off errors, stopping criteria and embedded iterative procedures, optimization issues for matrix polynomials, fast matrix computations, compressed/sparse sensing, PDE-constrained optimization, and applications and real time optimization.

The invited speakers are Larry Biegler (Carnegie Mellon), Nick Higham (University of Manchester), Adrian Lewis (Cornell University), Volker Mehrmann (Technische Universität Berlin), Mike Saunders (Stanford University), Valeria Simoncini (Università di Bologna), Jared Tanner (University of Edinburgh), and Andy Wathen (University of Oxford).

The Organising Committee is Roy Mathias, co-chair (University of Birmingham), Michal Kočvara, co-chair (University of Birmingham), Iain Duff (Rutherford Appleton Laboratory), Nick Gould, (Rutherford Appleton Laboratory), Daniel Loghin (University of Birmingham), Alastair Spence (University of Bath), Zdeněk Strakoš, (Charles University, Prague) and Philippe Toint (University of Namur).

For more information such as how to propose a mini symposium, submit an abstract and register, visit the conference website http://www.ima.org.uk/Conferences/2nd_numerical_linear_algebra.html.

Aston Webb Building, University of Birmingham

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880pp; ISBN: 978-981-256-087-2, US$109; ISBN: 978-981-256-132-9(pbk), US$59

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Spectral Graph Theory Workshop Participants

REPORTS ON CONFERENCES AND WORKSHOPS

Workshop on Spectral Graph Theory With Applications to Computer Science, Combinatorial Optimization and Chemistry

Rio de Janeiro, BrazilDecember 1-4, 2008

Report by Vladimir Nikiforov

This workshop was organized by the Production Engineering Program of the Federal University of Rio de Janeiro. The sessions were held at the hospitable Military Institute of Engineering, next to the famous mount Sugar Loaf. The meeting was sponsored and funded by a number of prominent Brazilian organizations and businesses, whose generous contributions are acknowledged at the workshop website: http://www.sgt.pep.ufrj.br/~tegrio/.

The theory of graph spectra is now a well established field of research in mathematics and in several applied sciences like chemistry, computer science and operational research, to name a few. In the last few decades an astonishing body of work has been produced in this area. In recognition of this strong development, the workshop was organized as a truly international forum that was attended by 74 participants from all around the globe.

The workshop brought together leading researchers on graph spectra that presented state of the art results and discussed new challenges. The topics included also applications to computer science, combinatorial optimization, chemistry and other branches of science.

The members of the Scientific Committee, chaired by Dragos Cvetkovic, all renowned specialists in spectral graph theory, delivered 12 plenary presentations of one hour. There were also 9 plenary presentations of 30 min and other 28 short communications, split into two parallel sessions.The topics that received the most attention included graph energy, digraph spectra, eigenvalue bounds and the spectrum of the signless Laplacian.

With ample opportunities for discussions and communication, the meeting was truly rewarding for the participants, among whom were many young researchers. The organizing committee, chaired by Nair de Abreu, did a fantastic job: in a splendid setting, the meeting ran smoothly and efficiently from the registration to the final cocktail. There were several social events, all turned unforgettable by the warm Brazilian hospitability.

The results presented at the workshop will appear, subject to refereeing and approval, in a special volume of Linear Algebra and Its Applications.


15th Haifa Matrix Theory ConferenceMay 18-21, 2009

Report by Raphael Loewy

The 2009 Haifa Matrix Theory Conference took place at the Technion-Israel Institute of Technology in Haifa, Israel, on May 18-21, 2009. This was the 15th in a sequence of matrix theory conferences held at the Technion since 1984.

More than 80 people participated in the conference, and 59 talks were presented. As in the previous meetings, talks were of 30 minutes duration and covered a wide range of topics related to matrix theory.

All talks on the third day of the conference were dedicated to Professor Leiba Rodman on the occasion of his 60th

birthday. This special session began with a warm greeting and appreciation of his work by Israel Gohberg, and was followed by 4 talks. He completed his doctoral studies under Professor Gohberg in 1978.

Professor Rodman works in the general areas of operator theory and matrix analysis. His current research interests include matrix and operator valued functions, linear control systems, differential equations, optimization and interpolation, combinatorial matrix theory, and applications such as control systems and signal processing.

He is the author or co-author of 7 books and about 290 papers. He is Co-Editor-in-Chief of Operators and Matrices and a member of the editorial boards of other journals, including Linear Algebra and Its Applications, Mathematical

Inequalities and Applications, Integral Equations and Operator Theory, and Complex Analysis and Operator Theory. He has also been Co-Editor of 7 special volumes in operator theory, matrix analysis and related topics.

One of the topics discussed in the conference was “Teaching of Linear Algebra”. Steve Leon described developments in linear algebra education reform, while Gilbert Strang spoke about key points in a linear algebra course. In addition to these plenary talks there was a special session in which three lecturers from the Technion described several initiatives aimed at making the teaching of linear algebra more interesting and more effective. These included a model for evaluating students’ understanding of the concept of equivalence relation, the use of clickers to improve interaction between students and teachers, and the matrix theory behind Google’s PageRank algorithm, which can be a good motivating introduction to a matrix theory course.

The conference was supported and held under the auspices of the Center for Mathematical Sciences at the Technion. The education lectures were supported by the Ruth and Allen Ziegler Center for Learning Sciences at the Technion’s Department of Education.

More information about the conference and photos can be found at its website http://www.math.technion.ac.il/cms/decade_2001-2010/year_2008-2009/matrix/.


Summer School and Advanced Workshop onTrends and Developments in Linear AlgebraInternational Centre for Theoretical Physics, Trieste, Italy, June 22-July 3, 2009

Report by Peter Šemrl

A Summer School and Advanced Workshop on Trends and Developments in Linear Algebra was held at The International Centre for Theoretical Physics (ICTP) in Trieste, Italy, June 22-July 10, 2009.

This event was organized by R. Bhatia, J. Holbrook, and S. Serra Capizzano, with the help of R.T. Ramakrishnan (who works full time at ICTP). During the first two weeks, a summer school was held on “Trends and Developments in Linear Algebra”, followed by a one week workshop at the Adriatico Guest House of the ICTP located only a few meters from the Adriatic sea, and next to the beautiful castle Miramare.

Three broad topics were chosen for the program: Matrix Analysis, Matrix Computations, and Quantum Information Theory. During the first two weeks each morning there were three lectures of 60 minutes. After a lunch break there was one more lecture followed by three problem/tutorial sessions.

Besides the organizers, the lecturers at the summer school were L. Elden, R. Horn, D. Kribs, M.B. Ruskai, P. Šemrl, and V. Simoncini. Some of the lecturers prepared lecture notes. Links to these can be found in the programme (http://cdsagenda5.ictp.trieste.it/full_display.php?smr=0&ida=a08167).

J.A. Antezana, J.S. Aujla, and I. Wrobel acted as tutors to help the lecturers with afternoon problem/tutorial sessions.

There were over 70 participants coming from Algeria, Argentina, Azerbaijan, Cameroon, Canada, China, Cuba, Egypt, Georgia, Hungary, India, Iran, Italy, Japan, Korea, Macedonia, Morocco, Nepal, Nigeria, Pakistan, Portugal, Romania, Senegal, Serbia, Singapore, Slovenia, Sudan, Tunisia, Ukraine, United Arab Emirates, United Kingdom, USA, and Vietnam. The lecturers agreed that it was a great joy to work with the enthusiastic participants. It is clear that some of them are very talented and will soon become successful mathematicians.

The workshop speakers were K. Audenaert, J.-Z. Bai, F. Benatti, G. Benenti, D. Bini, H.R. Chan, N. Guglielmi, T. Huckle, R. Kannan, H. Kosaki, J. Lawson, Y. Lim, V. Mehrmann, B. Meini, M. Moakher, D. Petz, P. Šemrl, E. Tyrtyshnikov, J. Zemanek, and X. Zhan. It was a special pleasure to attend shorter talks given by younger colleagues

J.A. Antezana, J.S. Aujla, B. Iannazzo, C. Manni, and I. Wrobel, who presented their research results of high quality in a very professional way.

The International Centre for Theoretical Physics in Trieste, Italy was founded by Abdus Salam in 1964. There he instituted the famous “Associateships” which make it possible for talented young scientists from developing countries to spend their vacations at ICTP in close touch with leading experts from all over the world and then return to their own countries for nine months of the academic year.

Abdus Salam was a Nobel Laureate in physics in 1979, together with Americans Steven Weinberg and Sheldon Glashow.

He was born in 1926 in Jhang, a small town in what is now Pakistan. After graduating from the University of Punjab he continued his studies in Cambridge, obtaining a PhD in theoretical physics in 1951. He then returned to Pakistan with the intention of founding a high

level research school at Punjab University. This turned out to be a difficult mission. The only possible way to continue his outstanding career in theoretical physics was to leave his own country and to work abroad.

Many years later, in 1964, he founded the International Centre for Theoretical Physics in Trieste. Salam died in 1996 after a long illness.

Today ICTP operates under a tripartite agreement among the Italian Government and two United Nations Agencies, UNESCO and IAEA. Its mission is to support the best possible science with special attention to the needs of developing countries. In particular, high-level training courses, workshops, conferences and topical meetings take place throughout the year at ICTP. The participation of scientists from developing countries is fully supported by ICTP. For more information, visit http://www.ictp.trieste.it/.

Abdus Salam


International Workshop on Scientific Computing and its ApplicationsShanghai, China, June 28-30, 2009

Report by Yimin Wei and Fuzhen Zhang

The International Workshop on Scientific Computing and its Applications was held in Shanghai Fudan University, China, from June 28 to 30, 2009. This meeting was dedicated to numerical analysis and related areas. About 30 mathematicians, including some graduate students, attended the conference.

The speakers were:Wei-wei SUN: Mathematical Modeling, Computation and Analysis for Heat and Moisture Transport in Fibrous MaterialsFu-zhen ZHANG: Recent Results and Research Problems of Hua MatricesJian-feng CAI: A Singular Value Thresholding Algorithm for Matrix CompletionMoody T. CHU: Data Mining and Applied Linear AlgebraHong-gang ZENG: Matrix Computation in Numerical Polynomial Algebra and Algebraic Geometry Chi-Tien LIN: Numerical Study of Solution to a p-System with Nonlinear DampingChun-hua OU: Traveling Waves of Non-local Reaction Diffusion EquationsXu-zhou CHEN: Discussion on Dimensionality Reduction and Data RepresentationRong-qing JIA: Riesz Bases of Wavelets and Applications to Numerical Solutions of Elliptic EquationsSan-zheng QIAO: The LLL Algorithm and Sphere Decoding Zi-cai LI: New Simple Local Refinements of Finite Element Method for a Corner Li-hong ZHI: Determining Singular Solutions of Polynomial Systems via Symbolic-Numeric Reduction to Geometric Involutive FormXiao-qing JIN: Tri-diagonal Preconditioner for Family of Toeplitz Systems with Financial ApplicationsZhong-xiao JIA: A Refined Harmonic Lanczos Bidiagonalization Method and an Implicitly Restarted Algorithm for Computing the Smallest Singular Triplets of Large MatricesMu-sheng WEI: A Canonical Decomposition of a Right Invertible System with Applications Hai-wei SUN: Shift-Invert Arnoldi Approximation to a Toeplitz Matrix Li QIU: Measure of Instability The event was organized by Yimin Wei of Fudan University and sponsored by the Shanghai Key Laboratory of Contemporary Applied Mathematics of Fudan University, Shanghai Municipal Science and Technology Committee, Shanghai Municipal Education Committee (Dawn Project), and the National Natural Science Foundation of China.

Participants, International Workshop on Scientific Computing and its Applications

ORDER ONLINE: www.siam.org/catalogOr use your credit card (AMEX, MasterCard, and VISA): Call SIAM Customer Service at +1-215-382-9800 worldwide · Fax: +1-215-386-7999 E-mail: [email protected]. Send check or money order in US dollars to: SIAM, Dept. BKIL09, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Members and customers outside North America can also order SIAM books through Cambridge University Press at www.cambridge.org/siam.

Numerical Matrix Analysis: LinearSystems and Least SquaresIlse C. F. IpsenThis self-contained textbook presentsmatrix analysis in the context of numericalcomputation with numerical conditioningof problems and numerical stability ofalgorithms at the forefront. Using a uniquecombination of numerical insight and

mathematical rigor, it advances readers’ understanding of twophenomena: sensitivity of linear systems and least squaresproblems, and numerical stability of algorithms. The material ispresented at a basic level, emphasizing ideas and intuition, andeach chapter offers simple exercises for use in the classroom andmore challenging exercises for student practice.2009 · xiv + 128 pages · Softcover · ISBN 978-0-898716-76-4List Price $59.00 · SIAM Member Price $41.30 · OT113

Matrix PolynomialsI. Gohberg, P. Lancaster, and L. RodmanClassics in Applied Mathematics 58

A comprehensive treatment of the theory of polynomials in acomplex variable with matrix coefficients. Basic matrix theorycan be viewed as the study of the special case of polynomials offirst degree; the theory developed in Matrix Polynomials is anatural extension of this case to polynomials of higher degree. It has applications in many areas, such as differential equations,systems theory, the Wiener–Hopf technique, mechanics andvibrations, and numerical analysis. 2009 · xxiv + 409 pages · Softcover · ISBN 978-0-898716-81-8List Price $92.00 · SIAM Member Price $64.40 · CL58

Functions of Matrices: Theory and ComputationNicholas J. HighamThe only book devoted exclusively to matrix functions, thisresearch monograph gives a thorough treatment of the theory ofmatrix functions and numerical methods for computing them. Theauthor’s elegant presentation focuses on the equivalent definitionsof f(A) via the Jordan canonical form, polynomial interpolation,and the Cauchy integral formula, and features an emphasis onresults of practical interest and an extensive collection of problemsand solutions. This book is for specialists as well as anyone wishingto learn about the theory of matrix functions. 2008 · xx + 425 pages · Hardcover · ISBN 978-0-898716-46-7List Price $59.00 · SIAM Member Price $41.30 · OT104

APPLIED MATHEMATICS

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Assignment ProblemsRainer Burkard, Mauro Dell'Amico, and Silvano MartelloA comprehensive treatment of assignment problems from theirconceptual beginnings in the 1920s through present-daytheoretical, algorithmic, and practical developments. The authorshave organized the book into 10 self-contained chapters to makeit easy for readers to use the specific chapters of interest to themwithout having to read the book linearly. This book is a usefultool for researchers, practitioners, and graduate students. 2009 · xx + 382 pages · Hardcover · ISBN 978-0-898716-63-4List Price $110.00 · SIAM Member Price $77.00 · OT106

Learning MATLABTobin A. DriscollThis engaging book is a concise introductionto the essentials of the MATLAB®

programming language and is ideal for readersseeking a focused and brief approach to thesoftware. Learning MATLAB containsnumerous examples and exercises involvingthe software’s most useful and sophisticated features and anoverview of the most common scientific computing tasks forwhich it can be used. The presentation is designed to guide newusers through the basics of interacting with and programming inthe MATLAB software, while also presenting some of its moreimportant and advanced techniques. 2009 · xiv + 97 pages · Softcover · ISBN 978-0-898716-83-2List Price $28.00 · SIAM Member Price $19.60 · OT115

Parallel MATLAB for Multicore and Multinode ComputersJeremy KepnerSoftware, Environments, and Tools 21

This is the first book on parallel MATLAB® and the first parallelcomputing book focused on the design, code, debug, and testtechniques required to quickly produce well-performing parallelprograms. MATLAB is currently the dominant language oftechnical computing with one million users worldwide, many ofwhom can benefit from the increased power offered byinexpensive multicore and multinode parallel computers.MATLAB is an ideal environment for learning about parallelcomputing, allowing the user to focus on parallel algorithmsinstead of the details of implementation.2009 · xxvi + 253 pages · Hardcover · ISBN 978-0-898716-73-3List Price $65.00 · SIAM Member Price $45.50 · SE21

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10th SIAM Applied Linear Algebra Conference, Monterey CA, Oct. 26-29, 2009

Report by Chen Greif

The 10th SIAM Conference on Applied Linear Algebra took place in Monterey, California, on Oct. 26-29, 2009. It was a record breaking meeting: over 200 minisymposia talks (almost double compared to the 2006 meeting!), along with approximately 80 contributed talks and 11 plenary talks, brought the overall number of presentations beyond 300. The participants were certainly spared of the need to think about how to fill their day. Talks went on from 8 a.m. to as late as 10 p.m. in two of the evenings, featuring no less than six parallel sessions. And there were plenty of other activities: a SIAG/LA business lunch, a conference banquet, a forward looking panel discussion, and much more.

Over 250 participants braved the harsh California weather conditions of approximately 20 degrees (Celsius) and beautiful sunshine, and spent the week at the glamorous Monterey Bay of California, located about 100 miles south of San Francisco. The Embassy Hotel, where the meeting took place, was pleasant, and the facilities were excellent.

The plenary talks were remarkably diverse and of a very high quality. The speakers, in order of their appearance, were: Andy Wathen, Oxford (preconditioners for PDE-constrained optimization), Michele Benzi, Emory (matrix functions), Shaun Fallat, Regina (ILAS speaker - matrix factorizations and total positivity), Karen Wilcox, MIT (model reduction), Françoise Tisseur, Manchester (quadratic eigenvalue problems), Christian Mehl, TU Berlin (ILAS speaker - indefinite linear algebra), Hugo Zaragoza, Yahoo! (link graphs), Michael Ng, Hong Kong (data mining and image processing), Zlatko Drmač, Zagreb (SIAG/LA Best Paper Prize - see more below), Dianne O’Leary, Maryland (confidence in image reconstruction), Michael Friedlander, UBC (sparse optimization), and Dan Spielman, MIT (combinatorial solution of systems with M-matrices).

There were two history sessions, featuring four speakers in each, over the course of two evenings. Drawing lessons from the previous meeting in Dusseldorf, where those talks were a huge draw but were scheduled in parallel with other talks, this time the history sessions stood alone as single sessions. They were very well attended despite the late hour of 8-10 p.m.

The SIAG/LA business lunch meeting was led by the activity group’s Chair, Daniel Szyld, who updated the SIAG members with the state of affairs in the group. Daniel described the activities that the SIAG has been involved with (such as the summer school in Spain in 2007), some statistics about

the attendance of SIAG members in recent meetings, and the upcoming election of new officers. The SIAG/LA best paper prize was also announced: Z. Drmač and K. Veselić, for their paper, New Fast and Accurate Jacobi SVD Algorithm, published in SIMAX in 2008.

Just to make sure no time was wasted on activities not involving talking linear algebra, a panel on the role of linear algebra in industrial applications took place on the third day, over lunch time. The session was moderated by Mark Embree, along with the panelists Iain Duff, Roger Grimes, Volker Mehrmann, and Rosemary Renaut.

The conference banquet featured a choice of salmon or chicken, but no alcohol unless you wanted to add more to the already hefty dinner fee. (The hotel bar must have been grateful for that -- everybody flocked to the lobby after dinner to get their fix.) The after-dinner speaker, Beresford Parlett from Berkeley, did not spare words to share with us a few well kept secrets from his days as a graduate student. We would share those with you, but in the spirit of ”what happens in SIAM ALA stays in SIAM ALA”, let us leave it at this...

Altogether, it was a most enjoyable meeting, with excellent talks and a wonderful diversity of topics. Special thanks go to Esmond Ng, the meeting’s Chair, for successfully taking on the momentous task of planning and running the meeting. Daniel Szyld, the SIAG/LA Chair, also played an important role in assuring the success of the meeting.

The next SIAM Applied Linear Algebra conference is scheduled to take place in Valencia, Spain, on June 18-22, 2012. The local organizers are Rafael Bru and José Mas.

Christian Mehl, ILAS Speaker

“Definite versus Indefinite Linear Algebra”

Shaun Fallat, ILAS Speaker“Matrix Factorizations and

Total Positivity”


Israel Gohberg: August 23, 1928 - October 12, 2009

By Leiba Rodman, College of William and Mary, Williamsburg, VA 23188

With deep sadness I inform that Israel Gohberg passed away on Monday, October 12, 2009.

Israel Gohberg received his mathematical education in the Soviet Union. After holding several teaching and research positions in Moldova, Gohberg emigrated to Israel in 1974, where he was appointed Professor of Mathematics at Tel Aviv University and (later) incumbent of the Nathan and Lily Silver Chair in Mathematical Analysis and Operator Theory. He held also part time positions for many years at Weizmann Institute of Science (Israel), Vrije Universiteit, Amsterdam (The Netherlands), and the University of Maryland, College Park (USA).

Israel Gohberg leaves an enormous, lasting legacy. MathSciNet lists 573 items for him as of October 26, 2009, including 90 books (authored or edited). He supervised 40 Ph.D. students (I am one of them), many of whom went on to build successful careers in mathematics.

In 1978 he established Integral Equations and Operator Theory, a premier journal in the area of operator theory and its applications, and since then has been Editor-in-Chief of this journal. He also established Operator Theory: Advances and Applications, a book series published by Birkhauser Verlag, which has about 200 volumes in print so far.

Israel Gohberg received numerous honors and awards, including 6 honorary doctoral degrees from universities in Germany, Austria, Moldova, Romania, and Israel. He was a corresponding member of the Academy of Science of Moldova and Foreign Member of the Royal Netherlands Academy of Arts and Sciences.

Israel Gohberg was a great research mathematician, educator, and expositor. His visionary ideas and charismatic leadership inspired many in our research community. He will be dearly missed.

Ed. Note: Professor Rodman and other colleagues of Israel Gohberg are preparing an extensive tribute to him, which will appear in Linear Algebra and Its Applications.

OBITUARY

Israel Gohberg


ARTICLES

Addendum to “Linear Algebra in Iran”, IMAGE Issue 42

We should have mentioned in our history of linear algebra in Iran (among other self-imposed restrictions on the list of articles there) that we included only those papers in linear algebra that had primary designation 15xx in Mathematical Reviews. Otherwise, our list would have been almost twice as long, as pointed out to us by M. S. Moslehian. Mehdi Radjabalipour and Heydar Radjavi.

Finding John Francis Who Found QR Fifty Years Ago

By Frank Uhlig, AuburnThanks to the Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, [email protected]

Half a century ago, John Francis developed his QR algorithm to compute the eigenvalues of matrices. He submitted his first QR paper on October 29, 1959 to the Computer Journal and his second one on June 6, 1961. Both appeared in October 1961 without any modification. The referee of the first paper, Velvel Kahan, had held its publication back while trying to understand why the algorithm converged.

The first paper contains a proof of convergence, but the question of why QR converges has occupied mathematicians ever since. In the meantime, John expanded, implemented and tested his algorithm. His second paper explains shifts; in fact the one later named after Jim Wilkinson first appears there. The theory of implicit shifts and the Implicit Q Theorem also appear in paper 2. A wee bit later, Vera Kublanovskaya explored a QR type matrix eigenvalue algorithm, but without the implementation, tests and enhancements that Francis’ papers had already offered.

Once the two Francis papers had appeared in the Computer Journal of 1961/62, John left the field and was soon lost to and forgotten by the numerical analysis and computations community. Here is the story of how he was located recently and came to speak at a minisymposium in June 2009, in honor of him and his remarkable algorithm.

After his papers appeared, Francis’ algorithm became the mainstay of dense matrix eigenvalue and singular value computations, the latter through a genial trick of Golub and Kahan to represent John Francis in Glasgow, 2009

Minisymposium ‘The QR Algorithm, 50 Years Later,’ Glasgow, June 23-25, 2009One side of the auditorium. Recognizable from left: Volker Mehrmann, Vladimir Khazanov, David Watkins, Charlie van Loan, Daniel Kressner, Françoise Tisseur, Chris Paige, Ilse Ipsen, Martin Gutknecht, Beresford

Parlett, Andy Wathen, Chris Fuller, Nick Trefethen


the singular values of a matrix inside an equivalent bi-diagonal matrix and then to use Francis’ Implicit Q Theorem to shift implicitly to obtain the singular values.

Finding matrix eigenvalues had long been one of the holy grails of numerical analysis. In the era following Abel’s 1805 discovery that polynomial equations generally do not admit closed form solutions, more than 8,000 papers and well over 300 books have been published with ideas and algorithms to find the roots of polynomials numerically. Why polynomials? Because matrix eigenvalues were defined theoretically by Cauchy via the characteristic polynomial in much the same way as many of us still teach the eigenvalue concept in elementary linear algebra classes today. As matrix models arose for solving ODEs and PDEs, matrix eigenvalues came to the fore and needed reliable computational methods. None of the polynomial root methods fulfilled that need of applied mathematics.

And thus Francis’ QR was included in the turn of the century list of the ten most important algorithms of the 20th century by Dongarra and Sullivan in 2000. Adding Gaussian elimination and the Euclidean algorithm to that list of most important algorithms of all mankind places Francis’ achievement in the elite dozen category of human history. Yet in 2000, no living mathematician knew whether John Francis was alive or where he lived. None of the then working mathematicians had met him before he had vanished in the early 1960s.

Soon after Gene Golub and this author began to search for John Francis, completely independently. I had started a project to develop software and adapt algorithms for use by chemical and biological engineers. Within this project, I became aware of many an engineering equation, constant, and apparatus that was associated with a person’s name and I searched far and wide for usable records of the name-givers. Eigenvalues and John Francis’ role in their computation led me to discover his whereabouts and part of his history. Similarly, Gene used internet searches and suggestions from our collective memory to find John.

Looking back at history, it appears that John’s whereabouts were actually found through his own famed algorithm: In the early, Bay area garage days of google, its founders looked for ways to search the internet effectively. They contacted Gene at Stanford who advised them of the page rank algorithm. This uses the singular value decomposition and helps google sort web data. Of course the SVD algorithm is just a specialization of Francis’ QR to bi-diagonal matrices and it also uses Francis’ Implicit Q Theorem. Searching for `John Francis’ in google brought both Gene and me a treasure of information on John. I enquired at the Museum of Science and Industry in Manchester that holds the files of Ferranti Ltd where John was employed after 1961. The enquiries gave me John’s middle name and his birthdate before the communications were abruptly terminated at the museum’s archive end for `privacy’ concerns. But this sufficed to locate John’s public school, his short Cambridge University student stay, and - last but not least - a recent church petition signed in his full name in Hove on the English Channel coast. That gave me all the information I would use in the footnote on Francis in the chemical engineering numerics book. And I left it at that.

And then chance took over. At an Operator Theory meeting in Moscow in July 2007, I happened to mention the often lost name-givers of quoted scientific entities, even in our own times. And Gene blurted out: “What about Francis?’’ to which I answered with

From the left: Martin Gutknecht, Chris Paige, David Watkins, John Francis, Volker Mehrmann, Beresford Parlett, and Andy Wathen


John’s birthdate, his full name, schools/years attended, work at Ferranti etc. Gene was stunned. We, the only two who knew about John Francis had met! Gene invited me to join him in visiting with John on August 7, 2007, but I had to decline. We wondered what it would be like to write John Francis’ biography for the community. Andy Wathen had intended to go with Gene that day, but at the last minute he was otherwise detained and Gene drove the 100 miles from Oxford to the coast by himself. Gene and John spent a couple of hours together and talked. Six days later Gene wrote a short e-mail to a number of friends about the visit. I received a copy, but was busy with the new semester at Auburn and did not communicate with Gene again. Then in November I learnt of Gene’s death in Stanford.

Now I was stunned. Again, no living mathematician had met or seen John Francis. Besides, I was apparently the only one who knew John’s whereabouts. I vowed to complete the task of bringing John back into the numerical analysts’ realm, into our community and consciousness.

To meet with John, I detoured through London’s Gatwick airport in July 2008 and spent 4 nights at a B&B two blocks from where John lives. We met, talked, ate out, walked, swam in the Channel together and had an interview in which John reminisced on the times and ideas around QR and I tried to give impromptu insights into the subsequent uses and impact of John’s seminal work. On the night before the interview I had given John a list of items and dates from his life that I knew about, as well as a list of possible topics and questions for the interview. The next morning, before the actual interview, John seemed very upset about the personal data I had shown him. How had I obtained that information? My honest answer of ‘from the internet’ was unbelievable to him. He had no idea how google worked nor how his Implicit Q Theorem and Gene and Velvel’s application of it had given me this information. An hour and a half later John understood and we were friends.

The next day I mentioned that I might try to organize a meeting on the history and recent advances of QR and asked John if he would come. John is a very private, unassuming and quiet person. After a couple of minutes and a block or two of walking down the road, he answered ‘I might’. I thanked him and we parted. Now I was left to re-introduce John Francis to our community. I chose the British Isles and contacted friends who all unfortunately were too engaged in research, editing, conference preparations etc. to be able to help until I convinced Andy Wathen that we owed John and our students and research community a chance to meet.

With Andy’s help, the 23rd Biennial Conference on Numerical Analysis at Strathclyde University in Glasgow June 23-25, 2009 became the venue for a mini-symposium on ‘The QR Algorithm, 50 Years Later, its Genesis by John Francis, and Subsequent Developments’. More than 100 participants listened to John Francis’ opening talk. He was followed by David Watkins on the perennial question of why, titled ‘Understanding the QR Algorithm, Part X’. I followed with applications of the Implicit Q Theorem and the tridiagonal DQR algorithm and its On2 polynomial root finder. Talks on recent extensions of QR were given by Chris Fuller on the nature of the general orthogonal group and Householder factorizations, by Karen Braman on multishift QR, by Daniel Kressner on parallel versions of QR and QZ, by Luca Gemignani on fast QR, and by Volker Mehrmann on structure preserving QR like algorithms. Another section of the mini dealt with its history in talks by Vladimir Khazanov, stepping in for Vera Kublanovskaya whose health prevents travel, on Vera’s work from QR to AB for matrix pencils. Beresford Parlett traced the antecedents of QR and LR to much earlier work on differential equations from the 1930s and Martin Gutknecht elaborated on how Rutishauser, whose 1958 LR paper inspired both John and Vera to discover QR, might have found qd and LR.

A historical scientific paper by Gene Golub and Frank Uhlig appeared in time for the Glasgow conference in the IMA Journal of Numerical Analysis, 29 (2009), 467-485 on June 8, 2009.

Andy Wathen, (Jen Pestana, hidden), and John Francis at dinner in Glasgow, 2009

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Page 23Fall 2008: IMAGE 41

Symmetric Sudoku Matrices and Magic Squares By Richard William Farebrother

2. Symmetric Sudoku Matrices For m odd, Farebrother [2, 3] has introduced a family of nonlinear transformations that carries each column of an m×m matrix into a primary diagonal, each primary diagonal into a row, each row into a secondary diagonal, and each secondary diagonal into a column. Further, he has shown [2, p.30] that the ‘prescription matrices’ associated with these nonlinear transformations may be used to convert an arbitrary m×m ‘seed matrix’ B containing m2 distinct symbols into an m×m(generalised) Sudoku matrix by replacing the cell marked (i,j) in the prescription matrix by the m×m matrix P i-1BP1-j where Pis an m×m permutation matrix of full cycle length such as that which carries the mth row into the (m-1)th row, ..., the second row into the first row, and the first row into the mth row.

It is immediately apparent that any such m×m matrix will satisfy a natural generalisation to n = m2 of the six sets of n = 9 constraints employed by Bailey, Cameron & Connelley (henceforth BCC) [1, p.390] in their definition of a 9×9 ‘symmetric Sudoku matrix’:

Now a symmetric Sudoku solution is an arrangement of the symbols 1,...,9 in a 9×9 grid in such a way that each symbol occurs once in each row, column, subsquare, broken row, broken column, and location.

Typical examples of such 9×9 Sudoku matrices have been identified independently by BCC [1, p.391] and Farebrother [2, p.30]. Readers are invited to use the method outlined in Farebrother [2, 3] to construct the corresponding 25×25, 49×49, 81×81, ... matrices using sets of 25, 49, 81, ... distinct symbols. They may then check whether the matrices so obtained are ‘orthogonal’ to their transposes in the sense understood in combinatorics. That is, whether, for each symbol σ in the original matrix and for each symbol τ in its transpose, the ordered combination (σ,τ) occurs in only one location. If so, then such matrices may be employed as the basis of Graeco-Latin squares. Farebrother & Styan [4, p.23] have shown that this is true of the 9×9 Sudoku matrices identified above.

2. Magic Squares The structures of such m2×m2 Sudoku matrices are particularly interesting when the symbols in the m×m seed matrix B are replaced by a sequence of numbers chosen in such a way that the elements in each of the rows, columns and primary or secondary cyclic-diagonals sum to the same fixed value.

A matrix with these properties may readily be obtained when m ≥ 5 by converting one of Farebrother's [3] m×m (‘bishop's move’) prescription matrices into a ‘knight's move’ matrix by interleaving the last (first) (m-1)/2 rows (columns) amongst the first (last) (m+1)/2 rows (columns) before identifying the typical element (i,j) with the value (i-1)m + j.

For instance, when m = 5, we have to interleave the last two rows of Farebrother's 5×5 prescription matrix [3, p.33] into the first three rows in the order 1, 4, 2, 5, 3:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

)5,2()3,4()1,1()4,3()2,5()4,1()2,3()5,5()3,2()1,4()3,5()1,2()4,4()2,1()5,3()2,4()5,1()3,3()1,5()1,2()1,3()4,5()2,2()5,4()3,1(

before identifying the typical element (i,j) with the value 5(i-1) + j to obtain the matrix with the required properties:

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

10181142241225816236192151751321911247203


Ancient Approximations To Pi

Richard William Farebrother

As noted in both solutions to Problem 36-1 (Image No.37, Fall 2006, pp.27–30), the simplest nontrivial solution in integers to the indeterminate equation x2-10y2 = 1 is x1 = 19, y1 = 6. These values serve to define the simple iterative procedure:

xn+1 = 19xn + 60yn yn+1 =6xn + 19yn

which gives rise to two sequences of ever closer approximations to the square root of 10, namely x1/y1 = 19/6= 3.1667, x2/y2 = 721/228 = 3.1623, ... and 10y1/x1 = 60/19 = 3.1579, 10y2/x2 = 2280/721 = 3.1623, ...

Although, strictly speaking, these expressions represent rational approximations to the square root of 10, we may follow ancient tradition and use them as the basis of rational approximations to the value of p =3.1416. Restricting ourselves to the ratio 10y2/x2 = 2280/721, we obtain a familiar expression for p by approximating this ratio by 2288/728 = 22/7=3.1429. Alternatively, we may approximate 2280/721 = 2560/809.54 by 2560/810 to obtain a value of 4(8/9)2 =3.1605, employed in some of the calculations featured in the so-called Rhind Mathematical Papyrus; see Midonick [2] or Joseph [1] for details.

Readers will be surprised to learn that this last approximate of p is also to be found amongst the cake recipes given in some English and American cookery books where it appears in the form: ”pour the mixture into a square tin of width 8 inches or a circular tin of diameter 9 inches” although most chefs prefer the less accurate value implied by the statement: ”pour the mixture into a square tin of width 7 inches or a circular tin of diameter 8 inches”. Assuming, in each case, that the area of the square tin is roughly the same as that of the circular tin (so that both cakes have the same depth), we obtain 4(8/9)2 = 3.1605 and 4(7/8)2 =3.0625 as culinary approximations to the value of p.

References

[1] George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, Penguin Books, Harmondsworth, 1990. Second edition Princeton University Press, Princeton, N.J., 2000.

[2] Henrietta Midonick. The Treasury of Mathematics, Penguin Books, Harmondsworth, 1968.

11 Castle Road, Bayston Hill, Shrewsbury, England SY3 0NF

References

[1] R. A. Bailey, Peter J. Cameron, & Robert Connelly. Sudoku, gerechte designs, revolutions, affine space, spreads, reguli, and Hamming codes. The American Mathematical Monthly, 115: (2008) 383--404.

[2] Richard William Farebrother. A general matrix transformation: problem and solution. Image No. 37, Fall 2006, p.39 and No.39, Fall 2007, pp.28--30.

[3] Richard William Farebrother. A simple procedure for a general matrix transformation: problem and solution. Image No.39, Fall 2007, p.32 and No.41, Fall 2008, pp.32--34.

[4] Richard William Farebrother & George P. H. Styan. Double Sudoku Graeco-Latin Squares. Image No. 41, Fall 2008, pp.22--24.

11 Castle Road, Bayston Hill, Shrewsbury, England SY3 0NF



Figure 2: Group picture taken at the ”German-Polish Workshopfor Young Researchers in Applied and Numerical Linear Alge-bra”, Mathematical Research and Conference Center in Bedlewo,Poland, February 2–5, 2006.

Figure 1: Pierre-Antoine Absil giving the Stiefel Centennial Lec-ture (left), Urs Hochstrasser (who coded the CG algorithm on aZ4 in 1952/54) and Eduard Stiefel Jr. during the “Apero” at theGAMM-ANLA workshop 2009 in Zurich. Pictures courtesy ofSimon Gutknecht.

The 2010 meeting also features a so called Young Researchers’Minisymposia on “Numerical Linear Algebra and Optimization”organized by the GAMM-ANLA members Melina Freitag andMartin Stoll.

The activity group regularly contributes to the GAMMnewsletter and the scientific journal “GAMM-Mitteilungen”;both are distributed to more than 2000 GAMM members and tolibraries. In 2005–2006, the “GAMM-Mitteilungen” publishedtwo “Themenhefte” (special issues) on Applied and NumericalLinear Algebra, edited by Heike Faßbender, with invited surveyarticles covering a wide range of topics in the field. Despite theGerman name of the journal, all articles are in English!Support for Young Scientists. From its beginning, the activ-ity group was intended to establish a forum for information andidentification for young scientists in the field. As already men-tioned above, GAMM-ANLA strongly encourages young scien-tists to contribute talks to workshops and proposals for Young Re-searchers’ Minisymposia at GAMM annual meetings. In 2006,the activity group organized a “German-Polish Workshop forYoung Researchers in Applied and Numerical Linear Algebra” inBedlewo, Poland2, see Figure 2 for a group picture taken there.A very intense program with talks by selected speakers and ex-tensive poster sessions led to a stimulating atmosphere whichgave rise to countless discussions among the participants. Theprize for the best poster was awarded to Marta Markiewicz forher poster on ”Numerical Simulation of Quantum Dots”. It isplanned to organize a similar workshop or summer school in2011.Outlook. The future of GAMM-ANLA will see a continuationof current activities, with more pronounced efforts to increasethe involvement and identification of young researchers, not onlywith GAMM-ANLA, but also with the GAMM in general. Theactivity group also seeks to intensify its collaboration with ILAS.Suggestions and contributions are always most welcome!

2See http://www.icm.tu-bs.de/ag_numerik/bedlewo/.

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BOOK REVIEW

Matrix Mathematics: Theory, Facts, and Formulas

By Dennis S. Bernstein

Princeton University Press, Princeton, New Jersey, 2009, 1137 pages.

Reviewed by Helmut LütkepohlFlorence, Italy

This is the second edition of a book that first appeared in 2005. I did not take note of the first edition but according to the author it was less than two thirds of the present version. The previously included material is updated and new material is added in many parts of the new edition. It is a remarkable source of matrix results. I will put it on the shelf nearest to my desk so that I have quick access to it. The book is an impressive accomplishment by the author.

He has structured the content in 12 chapters. The first chapter reviews some basic terminology and mathematical background material such as functions, relations and graphs and even more basic things like integers, real and complex numbers. It also presents a large number of inequalities and identities. Chapter 2 contains basic concepts and results related to matrices. For example, it includes results on ranks, determinants, traces, products, inverses, partitioned matrices and the like. Also important mathematical concepts such as subspace, null space, range space, cones and convex hulls are covered.

In Chapter 3 many useful results on a range of special matrices are presented. Matrices covered include Hermitian, symmetric, skew-Hermitian, skew-symmetric, idempotent, nilpotent, normal, projector, reflector, Hankel and Toeplitz matrices plus many more. Also Lie algebras and groups are discussed in this chapter.

Chapter 4 is devoted to polynomial matrices or matrix polynomials and rational transfer functions and covers among many other things the Smith and Smith-McMillan forms. Matrix decompositions such as companion, Jordan, Schur and singular value decompositions and related results are treated in Chapter 5. Chapter 6 deals with generalized inverses. Kronecker and Schur or Hadamard products are considered in Chapter 7. Positive-semidefinite matrices are the topic of Chapter 8. Vector and matrix norms are discussed

in Chapter 9 and matrix derivatives are treated in Chapter 10. Matrix exponentials and stability theory follow in Chapter 11, while the final chapter deals with linear systems and control theory.

The dimensions of the book are impressive. The chapters cover 879 pages which gives an idea of the number of results summarized in this volume. There is also a breathtaking bibliography with 1540 items and information on where the references are cited in the text. Moreover, there is a detailed author index stretching out over more than 10 pages.

The book concludes with a most remarkable index of 161 pages. This index is extremely valuable for locating specific results. Clearly, not many people want to read through hundreds of pages of matrix results to find a specific property of a matrix, if they want to use the book as a reference volume. For them the index is of great help. It is detailed and informative at the same time by indicating precisely what can be found for a specific term on a particular page. For example, for a controllability matrix the index tells us that we find a theorem on a controllable pair on p. 815, the definition on p. 809, a result on the rank on p. 809 and Sylvester’s equation on p. 869. The book also has an excellent listing of special symbols at the beginning. Again this is a great help for someone wanting to use the book as a reference.

The style is very concise, as one would expect from a mathematics book. It lists definitions and results. The latter are often called facts while more important results are signified as theorems or propositions. There are also many short remarks and very occasional examples and exercises. Despite the length of the book, proofs are not given for all results but often a reference is given. There are also many results that I would classify as nontrivial for which no proofs or references are provided. In general, the style is very efficient. In other words, the number of facts or results per page is rather large. Given the size of this volume, the reader may infer from that what a valuable source of matrix results it contains. Indeed, for some of the topics I would not be able to name a more comprehensive collection of results. So there is no doubt that this is an enormously valuable reference book.

Are there any features that I do not like? For me some of the remarks are just too short and uninformative. Too often they just point to other results without telling me why. For example, following Fact 2.11.11, there is the remark “See Fact 6.5.6.” This kind of remark is not untypical. It has its value because it refers to a related result. Still a brief explanation would be helpful for me. In some cases, I did not find a result that is useful for me. For example, the collection


JOURNAL NEWS

List of Books on the ILAS-IIC Website A list list of books about linear algebra and its applications is maintained by Oskar Baksalary, the Book Editor for IMAGE. This list is stored on the ILAS-IIC website (http://www.ilasic.math.uregina.ca/iic/). When you find new titles that would be good to include, or informative reviews that would be appropriate to reference, please send those suggestions to Prof. Baksalary (baxx @amu.edu.pl).

Call for Papers:Special Issue of LAA on Tensors and

Multilinear Algebra

Submission Deadline: September 30, 2010

Tensors are increasingly ubiquitous in various areas of applied, computational, and industrial mathematics because multilinear algebra arises naturally in many contexts. The purpose of this special issue is to bring together three lines of research -- multilinear algebraic techniques in multilinear algebra, linear algebraic techniques in multilinear algebra, and multilinear algebraic techniques in linear algebra.

The issue will attract the attention of mathematicians to challenges and recent findings in the rapidly developing field of applied and numerical multilinear algebra. It is open to all papers with significant new results where either multilinear algebraic methods play an important role or new tools and problems of multilinear algebraic nature are presented. Survey papers that illustrate common themes across disciplines and application areas, and especially where multilinear algebraic or tensor techniques play a central role, are also welcomed.

Papers must meet the publication standards of Linear Algebra and Its Applications and will be refereed in the usual way. For details on submitting papers, see http://www.stanford.edu/group/multilinear/laa.html.

Editors for this issue are Shmuel Friedland, University of Illinois at Chicago; Tamara G. Kolda, Sandia National Laboratories; Lek-Heng Lim, University of California, Berkeley; and Eugene E. Tyrtyshnikov, Russian Academy of Sciences.

of results related to vector and matrix derivatives is not as comprehensive as in some other collections, although many important results are there. Also, the half-vectorization operator is not covered, although mentioned, and discrete time state-space models would be useful for me.

Given the impressive size of the book it is perhaps not too surprising that some results are not easy to locate, despite the features that make it easier, e.g., the excellent index. For example, I was expecting to find the fact that the rank of an idempotent matrix is equal to its trace in Section 3.12, Facts on Idempotent Matrices. I found it only in Section 5.8 entitled Facts on Inertia and later again in Chapter 6 on Generalized Inverses. Finally, I would have liked to see more proofs or at least easily accessible references to proofs.

In summary, the main use of this impressive volume is clearly that of a reference book and for that purpose it is excellent. In the preface the author mentions different potential uses of the book, the main one being as a reference. He clearly achieves this goal. I can enthusiastically recommend it to anyone who uses matrices. The author has to be applauded for the accomplishment of putting together this impressive volume.

QUICK NOTES ABOUT BOOKS

New Books

Numerical Matrix Analysis: Linear Systems and Least Squares, by Ilse C. F. Ipsen, SIAM, 2009. This textbook presents matrix analysis in the context of numerical computation with special focus on conditioning of problems and stability of algorithms.

Applications of Linear Algebra to DNA Microarrays, by Amir Niknejad and Shmuel Friedland, VDM Verlag Dr. Müller, 2009. The purpose of this monograph is to condense the information that arises in molecular biology in general, and in gene expression data embedded in DNA microarrays in particular, and to use approximation by low rank matrices to complete missing data.

Matrix Methods: Applied Linear Algebra, by Richard Bronson and Gabriel B. Costa, 3rd ed., 2009. This text is suitable for a one semester first linear algebra course or a two semester course when supplemented with outside reading. It offers a balance between theory and matrix techniques. An appendix discusses use of technology.


Call for PapersJournal of Algebra, Number Theory:

Advances and Applications The Scientific Advances Publishers, Allahabad, India, invite original research papers and critical survey articles in all areas of algebra, number theory and their applications for possible publication in their new journal, Journal of Algebra, Number Theory: Advances and Applications (JANTAA).

JANTAA is being published in two volumes annually, with two issues in each. To see the contents of Volume 1, issue 2, visit http://scientificadvances.org/journals4P5.htm. For details about submissions, please visit the website http://scientificadvances.org/journals.

Call for Papers: Special Issue of LAA on 1st Montreal Workshop on Idempotent and

Tropical MathematicsDeadline for Submissions: December 31, 2009

The 1st Montreal Workshop on Idempotent and Tropical Mathematics was held June 29-July 3, 2009, at the University of Montreal, Canada. A special issue of LAA will be devoted to papers within the scope of this workshop, Tropical Algebra and Mathematical Physics and their Applications, and Tropical Geometry and its Applications. The Editor-in-Chief for this special issue is Hans Schneider. Topics include: Idempotent analysis; Idempotent functional analysis; Tropical geometry and algebra; Tropical and idempotent convex geometry; Mathematics of idempotent semirings and their applications; Linear algebra over semirings; Quantization, dequantization, and related topics of mathematical physics.

The deadline for submission of papers is December 31, 2009. Submissions will be refereed according to LAA standards. Send to any of the following special editors, preferably as PDF files in email attachments:

Peter Butkovic, School of Mathematics, Univ. of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom ([email protected]).

Alexander Guterman, Faculty of Algebra, Dept. of Math and Mechanics, Moscow State Univ. GSP-1, 119991, Moscow, Russia ([email protected]).

Jean Jacques Loiseau, IRCCyN, UMR CNRS 6597, Ecole Centrale de Nantes, 1 rue de la Noe, BP 92101 44321 Nantes cedex 3, France ([email protected]).

William M. McEneaney, Dept. of Mechanical and Aerospace Engineering, MC 0411, Univ. of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0411, USA ([email protected]).

Edouard Wagneur, GERAD, HEC Montréal, 3000,chemin de la Côte-Sainte-Catherine Montréal (Québec) Canada, H3T 2A7 ([email protected]).

ILAS NEWS

ILAS President-VicePresident Annual ReportSubmitted July 10, 2009

Call for Papers: JP Journal of Algebra, Number Theory and Applications

The Pushpa Publishing House invites original research papers and critical survey articles for possible publication after peer review in its journal JP Journal of Algebra, Number

Theory and Applications (JPANTA). This journal provides a common forum for significant research both on theoretical and on applied aspects of current interest in algebra and number theory and innovative links between these and various fields of applications. It is published in three volumes annually, each having two issues. All articles are reviewed in Mathematical Reviews, MathSciNet and Zentralblatt für Mathematik. Published articles and information about submissions are available at http://pphmj.com/journals/jpanta.htm.

1. The following were elected in the ILAS 2008 elections to offices with terms that began on March 1, 2009 and end on February 29, 2012:

Secretary-Treasurer: Leslie HogbenBoard of Directors: Dario Bini and Shaun Fallat

The following continue in ILAS offices to which they were previously elected:Vice-President: Chi-Kwong Li (term ends February 28, 2010)

Board of Directors:Wayne Barrett (term ends February 28, 2010)Judi McDonald (term ends February 28, 2011)Andre Ran (term ends February 28, 2011)Bryan Shader (term ends February 28, 2010)


James R. Weaver continues his appointment as Outreach Director until February 28, 2011.

Shmuel Friedland and William Watkins completed three-year terms on the ILAS Board of Directors on February 28, 2009. We thank both for their valuable contributions as Board members; their willingness to serve ILAS is most appreciated.

On February 28, 2009, Jeff Stuart completed his most recent term as ILAS Secretary-Treasurer, a position he held since 2000. ILAS extends its deep appreciation to Jeff for his years of careful management of ILAS’s finances and membership records. In accordance with ILAS bylaws, Jeff will serve for one year as a voting member of the ILAS Board, until February 28, 2010.

We thank the members of the Nominating Committee - Rajendra Bhatia, Jane Day, Christian Mehl, Michael Neumann (Chair), and Michael Tsatsomeros - for their work on behalf of ILAS, and to all of the candidates that agreed to have their names stand for election.

2. The following ILAS-endorsed meetings have taken place since our last report:

5th International Workshop on Parallel Matrix Algorithms and Applications (PMAA’08), Neuchatel, Switzerland, June 20-22, 2008

9th Workshop on Numerical Ranges and Numerical Radii, Williamsburg, USA, July 19-21, 2008

International Workshop on Operator Theory and its Applications (IWOTA 2008), Williamsburg, USA, July 22-26, 2008

18th International Workshop on Matrices and Statistics, Smolenice Castle, Slovakia, June 23-27, 2009

3. ILAS has endorsed the following upcoming non-ILAS conferences of interest to ILAS members:

3rd International Workshop on Matrix Analysis and Applications, Zhejiang Forestry University, Hangzhou/Lin’An, China, July 9-13, 2009

3rd International Symposium on Positive Systems: Theory and Application (POSTA09), Valencia, Spain, September 2-4, 2009

Applied Linear Algebra Conference in honor of Hans Schneider, Novi Sad, Serbia, May 24-28, 2010

Coimbra Meeting on 0-1 Matrix Theory and Related Topics, Department of Mathematics, University of Coimbra, Coimbra, Portugal, June 17-19, 2010

10th Workshop on Numerical Ranges and Numerical Radii, Krakow, Poland, June 27-29, 2010

4. The following ILAS Lecture at a non-ILAS conference has been delivered since the last report:

Hans Schneider spoke at the XIXth International Workshop on Operator Theory and its Applications, College of William and Mary, Williamsburg, VA, USA, July 22-26, 2008.

5. The following ILAS conferences are scheduled:

16th ILAS Conference, Pisa, Italy, June 21-25, 2010. The Organizing Committee consists of: Michele Benzi, Avi Berman, Dario Bini (Chair), Luca Gemignani, Leslie Hogben, Steve Kirkland, Julio Moro, Ilya Spitkovsky, Françoise Tisseur, and Eugene Tyrtyshnikov.

17th ILAS Conference, Braunschweig, Germany, 2011 (contact person is Heike Fassbender).

6. The Electronic Journal of Linear Algebra (ELA) is now compiling its 18th volume. Its Editors-in-Chief are Ludwig Elsner and Danny Hershkowitz. Vol. 17 (2008) was ELA’s largest ever, containing 45 papers (698 pages, a 51% increase over Vol. 16). Vol. 18 (2009) contains 26 papers so far (301 pages). There are 83 pending papers at present. The acceptance rate is 47%.

7. IMAGE, the semi-annual newsletter of ILAS has Chief Editor Jane Day. Several Consulting Editors are responsible for specific parts: Fuzhen Zhang (Problems Section), Peter Semrl (History of Linear Algebra), Oskar Baksalary (Book Reviews), Steve Leon (Education) and James Weaver (Advertisements).

Each issue is posted on the ILAS-IIC website http://www.ilasic.math.uregina.ca/iic/IMAGE/ and these are searchable PDF files. About 150 printed copies of each issue are actually mailed, but most ILAS members have opted to read it online only. The due dates for posting online are June 1 and December 1 each year. As of June 1, 2009, 42 issues have appeared. With issue 41, Hans Joachim Werner completed his service as an Editor-in-Chief of IMAGE, a position he held since 2000. ILAS extends its thanks to Professor Werner for his contributions to the newsletter.

8. ILAS-NET is managed by Sarah Carnochan Naqvi. This year the ILAS-Net distribution list is being reconciled with the membership list to ensure that all members are receiving ILAS-Net messages. There are approximately 600 email addresses on the current list.

9. The primary site of the ILAS INFORMATION CENTER (IIC) is at the University of Regina. Mirror sites are located


Increase in ILAS DuesAfter careful consideration and extensive discussion, the ILAS Board of Directors has approved an increase to the annual dues, effective immediately. The new dues structure is as follows: $50 (US) per year, with a $10 (US) reduction for those members that choose not to have the print version of IMAGE mailed to them.

Note that complete PDF versions of all issues of IMAGE are available at http://www.ilasic.math.uregina.ca/iic/IMAGE/ and that each new issue is posted promptly.

The Board has also agreed that there will be no further dues increases for the next five years. In keeping with past practice, dues can be waived annually for students, postdoctoral fellows, and people undergoing financial hardship. Those who have already renewed their memberships at the old dues level will still have their memberships honoured.

The scope of ILAS activities has expanded considerably since the Society’s founding. Recent years have seen ILAS providing partial support for students to attend ILAS annual conferences, as well as support for an increasing number of ILAS Lectures at non-ILAS conferences. Additional costs are anticipated in order to provide adequate technical support

Nominations for 2009 ILAS Elections Submitted by Steve Kirkland, ILAS President

The Nominating Committee for the 2009 ILAS elections has completed its work.

Nominated for a three year term, beginning March 1, 2010, as ILAS Vice-President is Chi-Kwong Li.

Nominated for the two open three-year terms, beginning March 1, 2010, as “at-large” members of the ILAS Board of Directors are: Christian Mehl, Berlin, Germany; Naomi Shaked-Monderer, Haifa, Israel; Jorma Merikoski, Tampere, Finland; Tin-Yau Tam, Auburn, US.

According to ILAS By-Laws, additional nominations may be made by electronic or regular mail by any three members of ILAS; prior approval of the nominee is required. All such nominations should be sent to the chair of the Nominating Committee, Rajendra Bhatia <[email protected]>, by November 4, 2009. Ballots will be sent out towards the end of 2009.

Thanks to the nominees for agreeing to stand for election, and to the Nominating Committee for their important service to ILAS: Rajendra Bhatia (Chair), Dario Bini, Roger Horn, Tom Laffey, and Francoise Tisseur.

at the Technion, Temple University, the University of Chemnitz, and the University of Lisbon. This year, an online membership renewal form for ILAS is being developed. Also the Table of Contents of most IMAGE issues have been added in order to assist with searching the large IMAGE files.

10. The Linear Algebra Education Committee is chaired by Steve Leon. This committee published a survey about second courses in linear algebra in Image issue 41. There was a very limited response. The Education Committee needs to gather more information on this matter before it can write a report on the findings.

Linear algebra education was an area of emphasis at the May 2009 Haifa Matrix Theory Conference, with invited education talks by Gil Strang and Steve Leon as well as three contributed education talks. The Linear and Numerical Linear Algebra Conference to be held in August 2009 at Northern Illinois University will also feature linear algebra education as an area of emphasis. Linear algebra education minisymposia are being planned for the SIAM Applied Linear Algebra Conference in October 2009 and for the ILAS Annual Conference in Pisa, Italy in June 2010.

Respectfully submitted,Steve Kirkland, ILAS President, [email protected] Li, ILAS Vice-President, [email protected]

New Address for ILAS President Steve Kirkland

As of July 1, 2009, Steve Kirkland’s new postal and email addresses are:

Steve Kirkland Hamilton Institute National University of Ireland, Maynooth Co. Kildare Ireland email: [email protected]

for the editing and production of the Electronic Journal of Linear Algebra (ELA), which has grown dramatically and is now receiving recognition for its high quality. Thus, the Board has approved the dues increase with the goals of strengthening ILAS, and ensuring that the Society can continue to play an active role in the growth of linear algebra.

Steve Kirkland, PresidentChi-Kwong Li, Vice-PresidentLeslie Hogben, Secretary-Treasurer


ILAS Members Named SIAM Fellows

The Society for Industrial and Applied Mathematics has inaugurated a Fellows Program. This honor is bestowed for outstanding contributions to applied mathematics and computational science, and these first honorees will be recognized during the 2009 SIAM Annual Meeting in Denver, Colorado. There are 191 members in this first group, and ten members of ILAS are among them: Carl de Boor, University of Wisconsin, MadisonNicholas J. Higham, University of ManchesterThomas Kailath, Stanford UniversityPeter Lancaster, University of CalgaryCleve Moler, MathWorks, Inc. Michael L. Overton, Courant Institute of Mathematical Sciences, New York UniversitySeymour V. Parter, University of Wisconsin, MadisonG.W. Stewart, University of Maryland, College ParkGilbert Strang, Massachusetts Institute of TechnologyPaul M. Van Dooren, Université Catholique de Louvain

The full list can be found at http://fellows.siam.org.

Send News for IMAGE Issue 44 by April 1, 2010

All items of interest to the linear algebra community are welcome, including: Announcements and reports of conferences and workshops News about striking developments in linear algebra and its applications Honors and awards Articles on history Survey articles Books, websites, funding sources News about journals Employment and other funding opportunities Problems and solutions Transitions: new appointments, responsibilities, deaths Linear algebra education Possible new corporate sponsors

Send material to the appropriate Editor by April 1, 2010: Problems and solutions to Fuzhen Zhang ([email protected]). Book news and reviews to Oskar Baksalary ([email protected]). History of linear algebra to Peter Šemrl ([email protected]). Linear algebra education to Steve Leon ([email protected]).

Advertisements to Jim Weaver ([email protected]). All other material and ideas to Jane Day ([email protected]) Photos are welcome. Historical tidbits and other short items as well as longer articles are welcome.

If you wish to submit an article after April 1, let Jane Day know as soon as possible when to expect your item, and it may be possible to include it.

Send material in plain text, Word, or both Latex and PDF (Article.sty with no manual formatting is preferred). Send photos in JPG format. Issue 44 will be published on June 1, 2010.

EMPLOYMENTResearch Positions at SISTA: Signals,

Identification, System Theory and Automation Research Division, University of

Leuven, Belgium

The research division SISTA of the Department of Electrical Engineering (http://www.esat.kuleuven.be/scd/) is one of the largest research groups of the University of Leuven (<http://www.kuleuven.be/>).

Research activities involve numerical linear algebra, system identification and control, algorithms for numerical optimization and machine learning (support vector machines), and signal processing. Numerous applications in industrial process control, biomedical signal processing, bio-informatics, health decision support systems and telecommunications are studied.

SISTA is looking for well motivated researchers to start a 4 year PhD or for experienced postdocs to pursue research activities in these fields and application areas. On the website (http://www.esat.kuleuven.be/scd/), one can find a description of more than 20 different specific research topics, in which there are openings for PhDs and postdocs. SISTA can offer a competitive salary and a stimulating research environment, settled in the nice historic city of Leuven and its surroundings, within a university whose origins go back to 1425 (http://www.leuven.be/).

Electronic applications, including a CV, a list of publications, names of two possible references, and a brief description of your research interests and motivation, should be submitted to [email protected].

www.tandf.co.uk/journals/lama

Linear and Multilinear

AlgebraAlgebraINCREASE

IN PAGES IN 2010

ILAS Members’ Rate of $118/£72Linear and Multilinear Algebra publishes original research papers that advance the study of linear and multilinear algebra, or that apply the techniques of linear and multilinear algebra in other branches of mathematics and science.

Visit www.tandf.co.uk/journals/lama for the full aims and scope.

EDITORS-IN-CHIEF:Steve Kirkland, National University of Ireland, Maynooth, IrelandChi-Kwong Li, College of William and Mary, USA

To subscribe to Linear and Multilinear Algebra at the ILAS members’ rate please visit www.tandf.co.uk/journals/offer/glma-so.asp.

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GLMA_Advert_final2.indd 1 9/9/09 17:06:22

IMAGE 43: Fall 2009 Page 35

IMAGE Problem Corner: Old Problems With SolutionsWe present solutions to Problems 41-4 and 42-1 through 42-6. Seven new problems are on the back cover; solutions are invited.

Problem 41-4: Integer Matrices with Unit Determinants IIProposed by Richard William Farebrother, Bayston Hill, Shrewsbury, England, [email protected]

(i) For positive integer values of d, f and g the equation (f2 − dg2)2 + d(2fg)2 = (f2 + dg2)2 supplies sets of three integerssatisfying the generalised Pythagorean equation x2 + dy2 = z2. Outline a matrix proof of this result.

(ii) In IMAGE Problem 36-1 [see IMAGE 37 (Fall 2006), pp. 20-23 for the problem and its solutions by Farebrother and by Werner]we were concerned with the construction of a 2 × 2 matrix with a determinant of plus one, and thus with the solution ofPell’s Equation f2 − dg2 = +1 where f and g are nonzero integers and d is a fixed positive integer (though not a square).If instead we assume that the matrix has a determinant of minus one, then we are concerned with the solution of a variant ofPell’s Equation (which, for obvious reasons, we shall name Nell’s Equation) f2− dg2 = −1. Identify integer solutions to thisequation for certain choices of d. Use the result established in Part (i) to obtain the corresponding solutions to Pell’s Equation.

Solution 41-4 by Hans Joachim Werner, University of Bonn, Bonn, Germany, [email protected]

By using the product rule, taking determinants of the matrices on both sides of the following matrix identity1 0

1 1

f2 − dg2 2dg2

−2f2 f2 − dg2

1 0

1 1

=

f2 + dg2 2dg2

0 f2 + dg2

yields (f2−dg2)2 +d(2fg)2 = (f2 +dg2)2. So f2− dg2 = (f2−dg2)2, where f := f2 +dg2, g := (2fg)2 and d := d. It is nowclear that if (f, d, g) ∈ N3 solves Nell’s equation, i.e., if f2 − dg2 = −1, then (f , d, g) solves Pell’s equation, i.e., f − dg2 = 1.

Next, we are interested in positive integer solutions to Nell’s equation

f2 − dg2 = −1. (1)

Clearly, for any positive integer f , the triplet (f, d, g) ∈ N3 with d := f2 + 1 and g := 1 trivially satisfies eqn. (1). Therefore, welist in the following table only all those integers f between 1 and 1000 for which there also exists a nontrivial solution to eqn. (1).Our table contains for these f -values only all the nontrivial (positive integer) solutions (f, d, g) ∈ N3.

f 7 18 32 38 41 43 57 68 70 82 93 99 107 117 118 132 143 157 168

d 2 13 41 5 2 74 130 185 29 269 346 58 458 10 557 697 818 986 1129

g 5 5 5 17 29 5 5 5 13 5 5 13 5 37 5 5 5 5 5

f 182 182 193 207 218 232 239 239 243 251 257 268 268 268 282 293

d 53 1325 1490 1714 1901 2153 2 338 2362 218 2642 17 425 2873 3181 3434

g 25 5 5 5 5 5 169 13 5 17 5 65 13 5 5 5

f 307 318 327 332 343 357 368 378 382 393 407 408 418 432 437

d 3770 4045 370 4409 4706 5098 5417 85 5837 6178 6626 985 6989 7465 1130

g 5 5 17 5 5 5 5 41 5 5 5 13 5 5 13

f 443 443 457 468 482 493 500 507 515 518 532 540 543 557 568

d 314 7850 8354 8761 9293 9722 89 10282 26 10733 11321 1009 11794 12410 12905

g 25 5 5 5 5 5 53 5 101 5 5 17 5 5 5


f 577 582 593 606 607 616 618 632 643 657 668 682 682 682

d 1970 13549 14066 2173 14738 1313 15277 15977 16538 17266 17849 5 125 18605

g 13 5 5 13 5 17 5 5 5 5 5 305 61 5

f 693 707 718 732 743 746 757 768 775 776 782 793 800

d 19210 19994 20621 21433 22082 3293 22922 23593 3554 113 24461 25154 761

g 5 5 5 5 5 13 5 5 13 73 5 5 29

f 807 807 818 829 832 843 857 868 882 882 882 893

d 1042 26050 26765 2378 27689 28426 29378 30137 37 925 31117 31898

g 25 5 5 17 5 5 5 5 145 29 5 5

f 905 907 915 918 932 943 944 957 968 982 993

d 2834 32906 4954 33709 34745 35570 5273 36734 37481 38573 39442

g 17 5 13 5 5 5 13 5 5 5 5

Many results in this table can be explained as follows. If (f, d, g) ∈ N3 is a solution to eqn. (1), then, for each k ∈ N, the triplets

(f + kg2, d+ k(2f + kg2), g), (kg2 − f, d+ k(kg2 − 2f), g) (2)

also represent solutions to eqn. (1). For observe that if (f, d, g) ∈ N3 satisfies f2+1g2 = d, then (f+x)2+1

g2 = f2+1g2 + x(x+2f)

g2 ∈ Nif and only if x(x+2f)

g2 ∈ N, which in turn happens whenever (a) x = kg2 or (b) x = kg2 − 2f for some k ∈ N. In case (a),f + x = f + kg2, so

(f + x)2 + 1g2

=(f + kg2)2 + 1

g2=f2 + 1g2

+2fkg2 + k2g4

g2= d+ k(2f + kg2) ∈ N.

This shows that if (f, d, g) solves eqn. (1), then triplet (f + kg2, d + k(2f + kg2), g) also solves eqn. (1). Likewise, in case (b),f + x = kg2 − f , so

(f + x)2 + 1g2

=(kg2 − f)2 + 1

g2=f2 + 1g2

+k2g4 − 2fkg2

g2= d+ k(kg2 − 2f),

which shows that if (f, d, g) solves eqn. (1), then triplet (kg2 − f, d + k(kg2 − 2f), g) also solves eqn. (1). Since (7, 2, 5)represents a solution to Nell’s equation (with g = 5), all other solutions with g = 5 can be reproduced by means of this method.

In what follows, let DN denote the set of all positive integers d satisfying f2+1d ∈ N for some number f ∈ N, and let D denote

the set of all those positive integers d for which there exists a triplet (f, d, g) ∈ N3 representing a solution to Nell’s equation (1).Needless to say, D is a proper subset of DN. The elements of DN which are less than or equal to 200 are as follows: 2, 5, 10, 13, 17,25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 146, 149, 157, 169,170, 173, 178, 181, 185, 193, 194, 197. As mentioned before, not all of these integers can be expected to be also elements of D. So,of course, all listed squares fail. Some other non-squares, such as the integer 34, also fail. Astonishingly, 34 is the only non-squareelement of DN that is ≤ 100 and has this property. We conclude our answer to this interesting problem by listing in our final tablenontrivial solutions to equation (1) whose d-values are between 2 and 100.

f 18 70 32 182 99 29718 1068 43 378 500 5604d 13 29 41 53 58 61 73 74 85 89 97g 5 13 5 25 13 3805 125 5 41 53 569

Also solved by the proposer.


Problem 42-1: Square Root of a Product of Two Orthogonal ProjectorsProposed by Oskar Maria Baksalary, Adam Mickiewicz University, Poznan, Poland, [email protected] Gotz Trenkler, Technische Universitat Dortmund, Dortmund, Germany, [email protected]

Let P , Q be n× n orthogonal projectors (i.e., Hermitian idempotent matrices) such that PQ is nonzero. Find a square root of PQ.

Solution 42-1.1 by Johanns de Andrade Bezerra, Campina Grande, PB, Brazil, [email protected]

We prove a more general result.LetA andB be two positive semidefinite matrices of the same order. ThenAB is diagonalizable and has nonnegative eigenvalues

[1, Corollary 2.3]. If C2 = AB, then C is a square root of AB.Let λ1, λ2, . . . , λk be the distinct eigenvalues of AB. Then AB = λ1E1 + λ2E2 + · · ·+ λkEk, where

EiEj = 0, i 6= j, E2i = Ei, i = 1, . . . , k, E1 + E2 + · · ·+ Ek = I, rangeEi = ker(λiI −AB).

Thus, (AB)∗ = λ1E∗1 + λ2E

∗2 + · · ·+ λkE

∗k , and rangeE∗i = ker(λiI − (AB)∗). It follows that

(rangeE∗i )⊥ = (ker(λiI − (AB)∗))⊥ = kerEi = range(λiI −AB).

Therefore, C =√λ1E1 +

√λ2E2 + · · ·+

√λkEk, C2 = AB, with rangeEi = ker(λiI −AB) and kerEi = range(λiI −AB).

Reference[1] Y.-P. Hong and R.A. Horn, The Jordan canonical form of the product of a Hermitian matrix and a positive semidefinite matrix, Linear AlgebraAppl. 147(1991)373–386.

Solution 42-1.2 by by Richard William Farebrother, Bayston Hill, Shrewsbury, England, [email protected]

Let P have rank r and Q have rank s and let m satisfy max(r, s) ≤ m ≤ n, then the n × n idempotent Hermitian matrices Pand Q may be written in the form P = XX∗ and Q = Y Y ∗ where X is an n × m matrix with m − r zero columns satisfyingX∗X = diag{Ir, 0m−r} and where Y is an n×m matrix with m− s zero columns satisfying Y ∗Y = diag{Is, 0m−s}.

Let them×mmatrixX∗Y have singular value decompositionX∗Y = UDV ∗ where U and V arem×m orthonormal matricessatisfying U∗U = Im and V ∗V = Im and where D is a real m×m diagonal matrix with nonnegative elements on its diagonal andzeros elsewhere [as any negative or complex factors dj/|dj | can be absorbed into the jth columns of U or V ].

ClearlyR = XUV ∗Y ∗ is the required square root of PQ as it satisfiesR2 = XUV ∗Y ∗XUV ∗Y ∗ = XUV ∗V D∗U∗UV ∗Y ∗ =XUDV ∗Y ∗ = XX∗Y Y ∗ = PQ.

Also solved by Eugene A. Herman, Hans J. Werner, and the proposers.

Problem 42-2: Properties of a Certain Matrix ProductProposed by Richard William Farebrother, Bayston Hill, Shrewsbury, England, [email protected]

Given an n×n nonzero matrixA with complex elements. Suppose thatB is a second n×nmatrix with complex elements satisfyingAB = BA and ABA = A. Investigate the properties of the n× n matrix C = BAB.

Solution 42-2.1 by Oskar Maria Baksalary, Adam Mickiewicz University, Poznan, Poland, [email protected] Gotz Trenkler, Dortmund University of Technology, Dortmund, Germany, [email protected]

The solution to the problem is given in what follows.Proposition. Let A,B ∈ Cn,n be such that AB = BA and ABA = A. Then C = BAB is the group inverse of A.Proof. The assertion will be established by straightforward verifications of the three conditions constituting the definition of thegroup inverse. Recall that the group inverse of A ∈ Cn,n is the unique matrix A# ∈ Cn,n satisfying the equations

AA#A = A, A#AA# = A#, AA# = A#A; (3)

see e.g., Section 4.4 in Ben-Israel & Greville (2003).On account of ABA = A, we have

ACA = ABABA = ABA = A

andCAC = BABABAB = BABAB = BAB = C,

what shows that C is a reflexive inverse of A, satisfying the first two conditions in (3).


The satisfaction of the third conditions in (3) is implied by AB = BA, which ensures that

CA = BABA = ABAB = AC.

Thus, C = A#. 2

Reference

A. Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications (2nd ed.), Springer-Verlag, New York, 2003.

Solution 42-2.2 by Susana Furtado, Faculdade de Economia do Porto, Portugal, [email protected] Maria Isabel Bueno, University of California, Santa Barbara, USA, [email protected]

Theorem. Let A,B ∈ Cn×n. If AB = BA and ABA = A then A, B and BAB are simultaneously similar to matrices of the forms0p ⊕ JA, B11 ⊕ J−1

A and 0⊕ J−1A , respectively, where 0 ≤ p ≤ n, JA ∈ C(n−p)×(n−p) is nonsingular and B11 ∈ Cp×p.

Proof. If A is nonsingular the result is trivial. Now suppose that A is singular, AB = BA and ABA = A. Let Q ∈ Cn×n be anonsingular matrix such that

A′ := QAQ−1 = J0 ⊕ JA,

where J0 ∈ Cp×p is nilpotent and JA ∈ C(n−p)×(n−p) is nonsingular. W.l.g., suppose that J0 = Jp1(0) ⊕ · · · ⊕ Jpk(0), where

Jpi(0) is the nilpotent Jordan block of size pi and p1 ≥ · · · ≥ pk > 0 (p = p1 + · · ·+ pk). Let

B′ := QBQ−1 =

B11 B12

B21 B22

,where B11 ∈ Cp×p. The equality AB = BA implies that JAB21 −B21J0 = 0 and J0B12 −B12JA = 0.

Since JA is nonsingular and J0 is nilpotent, B21 = 0 and B12 = 0. Therefore B′ = B11 ⊕ B22. Then, the equality ABA = Aimplies that B22 = J−1

A andJ0B11J0 = J0. (4)

If

B11 =

B′11 ∗

∗ ∗

,with B′11 ∈ Cp1×p1 , the equality AB = BA implies that Jp1(0)B′11 − B′11Jp1(0) = 0. If p1 > 1, a calculation shows that B′11 isupper triangular. From (4) it follows that

Jp1(0)B′11Jp1(0) = Jp1(0). (5)

Then p1 = 1 as, if p1 > 1, because B′11 would be upper triangular, the entry in position (1, 2) of the matrix in the left hand side of(5) would be 0 while the corresponding entry of the matrix in the right hand side is 1. Hence, p1 = · · · = pk = 1, i.e., J0 = 0. Then,

QAQ−1 = 0⊕ JA, QBQ−1 = B11 ⊕ J−1A and Q(BAB)Q−1 = 0⊕ J−1

A . 2

Remark. It is a simple consequence of the previous theorem that if AB = BA and ABA = A then BAB is the Drazin inverse ofA. We also note that if A and B are simultaneously similar to matrices of the forms described in the statement of the theorem thenAB = BA and ABA = A.

Also solved by Johanns de Andrade Bezerra, Eugene A. Herman, Hans J. Werner, and the proposer.

Problem 42-3: Commutators and Nonderogatory MatricesProposed by Geoffrey Goodson, Towson University, Towson, USA, [email protected]

Let A,B and D be n-by-n matrices. It is known that if AB − BA = D, where AD = DA, then D is nilpotent; and if A isdiagonalizable, then D = 0. If instead AB −BAT = D and AD = DAT , then D need not be nilpotent, but it is also known that ifA is diagonalizable then D = 0.

More generally, show that if AB − BAT = D and AD = DAT , where A is nonderogatory, then D is symmetric, singular andthe eigenvalue λ = 0 of D has a geometric multiplicity greater than or equal to the number of distinct eigenvalues of A.


Solution 42-3 by Johanns de Andrade Bezerra, Campina Grande, PB, Brazil, [email protected]

Let λi be any eigenvalue of A. For c ∈ C, with c 6= λi for any i, obviously, A − cI is non-singular, nonderogatory, (A − cI)B −B(A − cI)T = D ⇔ AB − BAT = D and (A − cI)D = D(A − cI)T ⇔ AD = DAT . Moreover, the number of distincteigenvalues of A equals the number of distinct eigenvalues of A− cI . Hence, for simplicity, we consider non-singular A.

Since A and AT are similar, it follows that A and AT have the same characteristic polynomials and minimal polynomials, andso AT is also nonderogatory; hence AT vi = λivi for some eigenvector of AT related with λi, and so DAT vi = ADvi = λiDvi.In fact, as AT is nonderogatory, that is, each eigenspace of AT has dimension equal to one, it follows that it is enough to show thatDvi = 0 to prove that D is singular and that the eigenvalue λ = 0 of D has a geometric multiplicity greater than or equal to thenumber of distinct eigenvalues of A.

Thus, let AB − BAT = D, then ABvi − BAT vi = ABvi − λiBvi = Dvi, hence λiABvi − λ2iBvi = λiDvi = ADvi =

A2Bvi − λiABvi, which implies A2Bvi − 2λiABvi + λ2iBvi = 0, and so (A− λiI)2Bvi = 0. If Bvi = 0, then Dvi = 0. Thus,

consider Bvi 6= 0 for some i. Suppose that (A− λiI)Bvi = Dvi 6= 0, then Bvi and (A− λiI)Bvi are linearly independent, and soDvi 6= kBvi, k constant, that is, Dvi−kBvi = xk, with xk 6= 0 (obviously, if xk = 0, thenDvi = 0 sinceDvi andBvi are linearlydependent). Hence,ADvi−kABvi = DAT vi−kABvi = λiDvi−kABvi = λiDvi−k(Dvi+λiBvi) = (λi−k)Dvi−λikBvi =Axk. Consider k = λi, and so xk = x, then −λ2

iBvi = Ax ⇒ −λ2iABvi = A2x = −λ2

i (Dvi + λiBvi) = −λ2i (x + 2λiBvi) =

−λ2i (x+ 2λi Ax−λ2

i) = −λ2

ix+ 2λiAx⇒ A2x− 2λiAx+ λ2ix = 0⇒ (A− λiI)2x = 0.

Let xk = Dvi − kBvi = ABvi − (λi + k)Bvi. Now consider k = −λi, and so xk = y, then y = Dvi + λiBvi = ABvi ⇒ADvi+λiABvi = DAT vi+λiABvi = λiDvi+λiy = Ay ⇒ λiADvi+λiAy = A2y = λ2

iDvi+λiAy ⇒ λi(Ay−λiy)+λiAy =A2y ⇒ (A− λiI)2y = 0.

y = sx, s constant, ⇔ Dvi + λiBvi = sDvi − sλiBvi ⇔ (1 − s)Dvi + (1 + s)λiBvi = 0 ⇔ Dvi = 1+ss−1λiBvi, but this

contradicts that Dvi 6= kBvi for any k ∈ C; so x and y are linearly independent, with s 6= ±1 (if s = ±1, then Dvi = 0).Since A is nonderogatory, it follows that dimker(A− λiI)2 = 2, and so if (A− λiI)x 6= 0, then (A− λiI)y = 0 since x and

y are linearly independent. Suppose that (A − λiI)x 6= 0, as (A − λiI)2x = 0, then (A − λiI)x 6= ax, a constant, which impliesAx− (λi + a)x = z = A(Dvi − λiBvi)− (λi + a)(Dvi − λiBvi)⇒

ADvi − λiABvi − (λi + a)Dvi + (λ2i + aλi)Bvi = z (6)

Let (A− λiI)y = 0⇒ (A− λiI)(Dvi + λiBvi) = 0⇒

ADvi + λiABvi − λiDvi − λ2iBvi = 0 (7)

Adding (6) and (7), we have 2ADvi−(2λi+a)Dvi+aλiBvi = z ⇒ 2λiDvi−(2λi+a)Dvi+aλiBvi = −aDvi+aλiBvi =z = −a(Dvi − λiBvi) = −ax, hence Ax − (λi + a)x = −ax, this yields (A − λiI)x = 0. However, since A is nonderogatory,it follows that dimker(A − λiI) = 1, and as x and y are linearly independent, there exists clearly a contradiction, and so thesupposition Dvi 6= 0 is false, and therefore Dvi = 0.

Now we will show that D is symmetric.Let D = AB −BAT and DT = BTAT −ABT , then

D −DT = A(B +BT )− (B +BT )AT = X, (8)

hence AD − ADT = DAT − ADT = DAT − (DAT )T = AX . Clearly, XT = −X and (AX)T = −AX = XTAT = −XAT ,and so

AX = XAT (9)

From equations (8) and (9), we can conclude that Xvi = 0.Let vi1, v

i2, . . . , v

iri

be a Jordan chain of AT corresponding to λi. If vi1 = vi 6= 0, then the following relations hold:

AT vi1 = λivi1 ⇒ XAT vi1 = λiXv

i1

AT vi2 − λivi2 = vi1 ⇒ XAT vi2 − λiXvi2 = Xvi1

AT vi3 − λivi3 = vi2 ⇒ XAT vi3 − λiXvi3 = Xvi2

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

AT viri− λiviri

= viri−1 ⇒ XAT viri− λiXviri

= Xviri−1

⇒

AXvi2 = λiXvi2

AXvi3 − λiXvi3 = Xvi2

. . . . . . . . . . . . . . . . . . . . . . . . . .

AXviri− λiXviri

= Xviri−1

.


Since A and AT are similar and nonderogatory, it follows that the Jordan chain of A and AT corresponding to λi have the samequantity of vectors. Hence, suppose that there exists a vector viri+1 such that

AXviri+1 − λiXviri+1 = Xviri. (10)

Thus, viri+1 = a1vi1 + · · ·+ ari

viri, a1, . . . , ari

∈ C, which implies Xviri+1 = a1Xvi1 + a2Xv

i2 + · · ·+ ari

Xviriand AXviri+1 =

a1AXvi1 + a2AXv

i2 + a3AXv

i3 + · · · + ariAXv

iri

, substituting in (10), we have that a2λiXvi2 + a3(Xvi2 + λiXv

i3) + · · · +

ari(Xviri−1 + λiXv

iri

)− λia2Xvi2 − λia3Xv

i3 − · · · − λiariXv

iri

= Xviri⇒ a3Xv

i2 + a4Xv

i3 + · · ·+ ariXv

iri−1 = Xviri

, butthis represents a contradiction since Xvi2, Xv

i3, . . . , Xv

iri

are linearly independent, hence Xvi2 = Xvi3 = · · · = Xviri= 0, and

therefore X = 0, that is, D is symmetric.

Also solved by the proposer.

Problem 42-4: Eigenvalue Interlacing for Normal MatricesProposed by Roger A. Horn, Department of Mathematics, University of Utah, Salt Lake City, Utah, USA, [email protected]

Prove the following generalizations of eigenvalue interlacing for bordered or additively perturbed Hermitian matrices.

1. Let n ≥ 3, A ∈Mn−1, x, y ∈ Cn−1, β ∈ C, and k ∈ {2, . . . , n− 1} be given and let

C =

A y

x∗ β

∈Mn.

Suppose that A and C are normal and let D ⊂C be a given closed disk that is not a point. If there are k eigenvalues of Ain D, then there are at least k − 1 eigenvalues of C in D. If there are p eigenvalues of C in D, then there are at least p − 2eigenvalues of A in D.

2. Let A ∈Mn and x, y ∈ Cn be given and let C = A+ yx∗. Suppose that A and C are normal and let D ⊂C be a given closeddisk that is not a point. Then the numbers of eigenvalues of A and C in D differ by no more than 1.

Solution 42-4 by the proposer Roger A. Horn, Department of Mathematics, University of Utah, Salt Lake City, Utah, USA,[email protected]

We invoke the following residual theorem to prove both claims.Theorem. Let C ∈ Mn be normal, let S ⊂ Cn be a given k-dimensional subspace, and let γ ∈ C and δ > 0 be given. Suppose that||Cx− γx|| ≤ δ for every unit vector x ∈ S. Then there are at least k eigenvalues of C in the disk {z ∈ C : |z − γ| ≤ δ}.Proof. Let λ1, . . . , λn be the eigenvalues of C. The eigenvalues of the positive semidefinite matrix G = (C − γI)∗(C − γI) are|λ1 − γ|2, . . . , |λn − γ|2. We have assumed that x∗Gx ≤ δ2x∗x for all x ∈ S, so at least k eigenvalues of G are not greater than δ2

[1, Theorem 4.3.21]. 2

For a stronger form of the residual theorem, see [2, Theorem 4.1].In the following, let {ξ1, . . . , ξk} be an orthonormal set of eigenvectors of a normal matrix A corresponding to its eigenvalues

λ1, . . . , λk in a closed disk D, which has center γ and radius δ > 0. Let E = span{ξ1, . . . , ξk} and check that z ∈ E ⇒‖(A− γI)z‖ ≤ δ ‖z‖ (write z as a linear combination of ξ1, . . . , ξk).

Proof of #1. We have dim E + dim{x⊥} = k + (n − 2) = (n − 1) + (k − 1), so the dimension of E ∩ {x⊥} is at least k − 1.Thus, the dimension of

S =

ζ =

z

0

∈ Cn : z ∈ E ∩ {x⊥}

is also at least k − 1. Then ζ ∈ S ⇒ ||(C − γI)ζ|| = ||(A− γI)z|| ≤ δ||z||. Apply the residual theorem.

Proof of #2. We have dim E + dim{x⊥} = k+ (n− 1) = n+ (k− 1), so the dimension of S = E ∩{x⊥} is at least k− 1. Thenz ∈ S ⇒ ‖(C − γI)z‖ = ‖(A− γI)z‖ ≤ δ||z||. Apply the residual theorem to conclude that if A has k eigenvalues in D, then Chas at least k − 1 eigenvalues there. Now interchange A and C in the preceding argument to show that if C has p eigenvalues in D,then A has at least p− 1 eigenvalues there. Conclude that k + 1 ≥ p ≥ k − 1.

References[1] R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.[2] R. Horn, N. Rhee, and W. So, Eigenvalue inequalities and equalities, Linear Algebra Appl. 270(1998)29-44.


Problem 42-5: Obtuse Basis and Acute Dual Basis for Euclidean SpaceProposed by Tin-Yau Tam, Auburn University, Auburn, USA, [email protected]

If {e1, . . . , en} is an obtuse basis of the Euclidean space E (i.e., (ei, ej) ≤ 0 for all i 6= j), prove that the dual basis {e∗1, . . . , e∗n}(i.e., (e∗i , ej) = δij) is acute (i.e., (e∗i , e

∗j ) ≥ 0 for all i 6= j).

Solution 42-5.1 by Eugene A. Herman, Grinnell College, Grinnell, IA, [email protected]

We prove the following equivalent statement: If {u1, . . . , un} and {v1, . . . , vn} are bases of Rn such that uj · vk = δjk for all j, kand uj ·uk ≤ 0 for all j 6= k, then vj ·vk ≥ 0 for all j 6= k. We also use another equivalent statement: If P is a positive definite n×nmatrix whose off-diagonal elements are nonpositive, then the off-diagonal elements of P−1 are nonnegative. Here is why these twostatements are equivalent. If {u1, . . . , un}, {v1, . . . , vn} are given as above, let P = U TU , where U = [u1 · · · un], and note thatP−1 = V TV , where V = [v1 · · · vn]. Conversely, if P is a given positive definite matrix, then P = U TU for some nonsingularmatrix U .

We give a proof by induction on n. The n = 1 case is trivial. For the n = 2 case we use the second of our two equivalentstatements: P−1 =

[a bb c

]−1 = 1ac−b2

[c −b−b a

]. Since P is positive definite, its determinant, ac − b2, is positive; hence the

conclusion follows.Now assumem ≥ 3 and that the statement is true for all n < m. By symmetry, to prove the first of our two equivalent statements,

we need only prove that v1 · v2 ≥ 0. We use the fact that Rm = Span{v1, v2} ⊕W , where W = Span{u3, . . . , um} and ⊕ denotesan orthogonal direct sum. Thus

u1 = av1 + bv2 + w for some a, b ∈ R, w ∈W (11)

Multiply equation (11) by v T2 , v T

1 , and u T2 , in turn:

0 = v2 · u1 = a(v1 · v2) + b|v2|2 ⇒ b = −a(v1 · v2)/|v2|2

1 = v1 · u1 = a|v1|2 + b(v1 · v2) = a|v1|2|v2|2 − (v1 · v2)2

|v2|2

and so a > 0 by Cauchy-Schwarz.

0 ≥ u2 · u1 = b+ u2 · w = −a(v1 · v2)|v2|2

+ u2 · w

and so

u2 · w ≤ a(v1 · v2)|v2|2

(12)

Now let P = ATA, where A = [u3 · · · um]. Since P is a (m − 2) × (m − 2) positive definite matrix whose off-diagonal entriesare nonpositive, the off-diagonal entries of P−1 = (ATA)−1 are nonnegative. The matrix A(ATA)−1AT projects orthogonally ontoW = Span{u3, . . . , um}, and so w = A(ATA)−1ATu1 by (1). Note that ATu1 has all nonpositive entries and thus so does x =(ATA)−1ATu1. Hencew = Ax = x3u3+ · · ·+xmum, where x3, . . . , xm are the entries of x. Therefore, since x3 ≤ 0, . . . , xm ≤ 0,

u2 · w = u2 · (x3u3 + · · ·+ xmum) = x3(u2 · u3) + · · ·+ xm(u2 · um) ≥ 0

By equation (12), we have v1 · v2 ≥ 0 since a > 0.

Solution 42-5.2 by the proposer Tin-Yau Tam, Auburn University, Auburn, AL, [email protected]

If {f1, . . . , fm} is a set of vectors in E such that (fi, fj) ≤ 0, i 6= j, then for any real scalars ai

(∑i

|ai|fi,∑i

|ai|fi) =∑i,j

|aiaj |(fi, fj) ≤∑i,j

aiaj(fi, fj) = (∑i

aifi,∑i

aifi).

Let {e∗1, . . . , e∗n} be the dual basis of {e1, . . . , en}. For each j = 1, . . . , n, write e∗j =∑i aiei since e1, . . . , en is a basis of E.

Consider −e∗j , e1, . . . , en, (m = n + 1) and the sum 0 = −ej +∑i aiei. So 0 = ej +

∑i |ai|ei and thus 0 =

∑i(ai + |ai|)ei.

Hence ai ≥ 0 for all i. Finally (e∗j , e∗k) = (

∑i aiei, e

∗k) = ak ≥ 0.

Solution 42-5.3 by Hans Joachim Werner, University of Bonn, Bonn, Germany: [email protected]

Let (V, 〈·, ·〉) be a Euclidean space of finite dimension n and let B = {v1, . . . , vn} be an ordered basis of V. The transformation

T : V → Rn

x 7→ T (x) =: x = (xi)i=1,2,...,n,


where x = (xi)i=1,2,...,n denotes the coordinate vector of x with respect to the given ordered basis B (i.e., x =∑ni=1 xivi), is an

isomorphism, i.e., T is linear, one-to-one and onto. We further note that the matrix

G := G(v1, v2, . . . , vn) :=

〈v1, v1〉〈v2, v1〉 · · · 〈vn, v1〉

〈v1, v2〉〈v2, v2〉 · · · 〈vn, v2〉...

.... . .

...

〈v1, vn〉〈v2, vn〉 · · · 〈vn, vn〉

is called the GRAMIAN MATRIX or the GRAM of the vectors v1, v2, . . . , vn. Since B is a basis for V, this Gramian matrix G ispositive definite and symmetric. In what follows, let B = {v1, v2, . . . , vn} be an obtuse basis for V and let B∗ = {w1, w2, . . . , wn}be the dual basis, i.e., let 〈wi, vj〉 = δij for all i, j = 1, 2, . . . , n. In this case, it is clear that G is a Z-MATRIX (i.e., a real matrixwith nonpositive off-diagonal elements) because B is an obtuse basis. We know that every positive definite symmetric Z-matrix isan M -MATRIX. So the implication Gx ≥ 0⇒ x ≥ 0 holds true. This implication is equivalent to G−1 ≥ 0, i.e., the nonsingularinverse G−1 of G is nonnegative. In order to complete our proof, it now suffices to show that

G−1 = (〈wi, wj〉)j,i=1,...,n .

For convenience, let wi denote the coordinate vector of the dual basis vector wi (i = 1, 2, . . . , n) with respect to the given orderedobtuse basis B = {v1, . . . , vn}. Notice that 〈wi, vj〉 = etjGwi, where ej denotes the jth UNIT COLUMN that contains a 1 in the jth

position and zeros everywhere else; observe that this unit vector ej is trivially the coordinate vector of vj with respect to B. LettingW := (w1, w2, . . . , wn) produces GW = In, where In is the identity matrix of order n, and thus W = G−1. Next notice that

(〈wi, wj〉)j,i=1,...,n = W tGW (13)

and substitute W = G−1 in (13). Because G−1 is a symmetric nonnegative matrix, it follows that (13) can be rewritten as

(〈wi, wj〉)j,i=1,...,n = G−1 ≥ 0.

This says that the dual basis is acute and so our solution is complete.

Problem 42-6: Positive Semidefinite MatricesProposed by Fuzhen Zhang, Nova Southeastern University, Fort Lauderdale, USA, [email protected]

Let a and b be (positive) real numbers such that a3 + b3 = 2. Show that a+ b ≤ 2. Discuss the analog of this for matrices: If A andB are n× n positive semidefinite matrices (i.e. A,B ≥ 0) such that A3 +B3 = 2In, does it necessarily follow that A+B ≤ 2In?

Solution 42-6.1 by Jan Hauke, Adam Mickiewicz University, Poznan, Poland, [email protected]

OBSERVATION 1. Let a, b, c, r be real positive, and r ≥ 1. Then ar + br = 2cr implies a+ b ≤ 2c.Proof. For r = 1 the result is trivial. For r > 1 due to the symmetric role of a and b we could assume that a ≥ b, which impliesc ≤ a < 21/rc. Let us analyze the function f(a) = a+ (2cr − ar)1/r. Observe that the derivative f ′(a) = 1− [2(c/a)r − 1]1−1/r]is negative for c < a < 21/rc. So, it is enough to note that the function f(a) has a maximum f(c) = 2c. 2

OBSERVATION 2. For positive semidefinite matrices A and B, Ar +Br = 2cI implies AB = BA.

Proof. Let the spectral decomposition of matrix B = UΛU∗. Then the spectral decomposition of A = UDU∗, withD = (2cI − Λr)1/r, and as a result the commutativity of A and B is obvious [a matrix function is understood as in [1, p. 526]. 2

OBSERVATION 3. For positive semidefinite matrices A and B, Ar = Br = 2cI implies A+B ≤ 2cI.Proof. By Observation 2 the commutativity of A and B implies that it is enough to compare diagonal elements of diagonal matricesin their spectral decompositions. Therefore by Observation 1 the result is obvious. 2

Remark. For r = 3 and c = 1 we obtain the version of Problem 42-6.

Reference[1] C.D. Meyer, Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics, Philadelphia, 2000.


Solution 42-6.2 by Eugene A. Herman, Grinnell College, Grinnell, IA, [email protected]

More generally, if a and b are non-negative real numbers such that ap + bp = 2 for some integer p > 1, one may readily show thata+ b ≤ 2. We show that if A and B are n× n positive semidefinite matrices such that Ap +Bp = 2In for some integer p > 1, thenA+B ≤ 2In.

A is unitarily similar to a diagonal matrix with non-negative entries along the diagonal. By applying the same similarity transfor-mation to B, we may assume that A is this diagonal matrix. Let E1, . . . , Em be the distinct eigenspaces of B and let λ1, . . . , λm bethe corresponding eigenvalues of B. Since B is positive semidefinite, we have E1⊕· · ·⊕Em = Cn, Ej ⊥ Ek whenever j 6= k, andλj ≥ 0 for j = 1, . . . ,m. Thus E1, . . . , Em are the eigenspaces of Bp, and the corresponding eigenvalues of Bp are λp1, . . . , λ

pm.

Since Ap + Bp = 2In and Ap is diagonal, Bp is diagonal as well, and the eigenspaces of Ap (and hence A) are also E1, . . . , Em.Let µ1, . . . , µm be the corresponding eigenvalues of A. If v ∈ Ej for some j, 1 ≤ j ≤ m, then

(A+B)(v) = (µj + λj)v, and 2v = (Ap +Bp)(v) = (µpj + λpj )(v)

Since µpj + λpj = 2 and µj , λj are non-negative, we know that µj + λj ≤ 2. Thus, A+B ≤ 2In.Note: The above proof extends to all real powers p > 1. Here we assume the usual definition of Mp for noninteger powers p

when M is Hermitian: For some unitary matrix U , we have U∗MU = diag(d1, . . . , dn); define Mp to be Udiag(d p1 , . . . , dpn)U∗.

Solution 42-6.3 by Zejun Huang and Xingzhi Zhan, East China Normal University, Shanghai, China, [email protected],[email protected]

We prove a more general result. We need the following lemma.Lemma. (Lowner-Heinz, [1]). If A ≥ B ≥ 0 and 0 ≤ r ≤ 1, then Ar ≥ Br.Theorem. Let A and B be positive semidefinite matrices of the same order such that Ak + Bk = tI, where k is a positive integer,t ≥ 2 is a real number and I is the identity matrix. Then A+B ≤ tI.Proof. The case k = 1 is trivial. Next we suppose k ≥ 2. First we prove that for real numbers t ≥ 2, we have

f(x) = (t− x)k + xk − t ≥ 0 for all x ≥ 0.

In fact, considering the derivative

f ′(x) = k[xk−1 − (t− x)k−1] < 0, if x < t/2; = 0, if x = t/2; > 0, if x > t/2,

and

f(t/2) =tk

2k−1− t = [(t/2)k−1 − 1]t ≥ 0,

we obtain f(x) ≥ 0. Hence

(tI −B)k +Bk − tI = f(B) ≥ 0, i.e., (tI −B)k ≥ tI −Bk = Ak.

Applying Lemma with r = 1/k we get tI −B ≥ A, i.e., A+B ≤ tI. 2

We remark that the theorem does not hold when t < 2 even for scalars. For a given integer k ≥ 2 and a positive real numbert < 2, let a = b = (t/2)1/k. Then ak + bk = t, but a+ b = 2(t/2)1/k > t.

Reference[1] X. Zhan, Matrix Inequalities, LNM 1790, Springer, Berlin, 2002.

Solution 42-6.4 by Minghua Lin, University of Regina, Saskatchewan, Canada, [email protected]

The first assertion follows from the power mean inequality a+b2 ≤ (a

3+b3

2 )13 , for a, b ≥ 0.

Since A3 + B3 = 2In, pre and post multiply the equality by A3 respectively, we see that A3B3 = B3A3, that is, A3 and B3

commutes. Since A and B are positive semidefinite matrices, then A and B commutes [1, Theorem 2.5.15]. Therefore, A,B aresimultaneously diagonalizable. There exists a unitary matrix U such that 2In = U∗(A3 +B3)U = diag(λ3

1(A) + λ3i1

(B), λ32(A) +

λ3i2

(B), · · · , λ3n(A) +λ3

in(B)). By the scalar case, we have U∗(A+B)U = diag(λ1(A) +λi1(B), λ2(A) +λi2(B), · · · , λn(A) +

λin(B)) ≤ diag(2, 2, · · · , 2) = 2In. This completes the proof.

Reference[1] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985.

Editorial note: The statements in fact hold for real numbers a, b, odd integer k in place of 3, and Hermitian matrices A, B.

Also solved by Oskar Maria Baksalary and Gotz Trenkler, Richard William Farebrother, Carols M. da Fonseca, Heinz Neudecker,Alicja Smoktunowicz, Hans J. Werner, Iwona Wrobel, and the proposer.


IMAGE Problem Corner: New Problems

Problems: We introduce 7 new problems in this issue and invite readers to submit solutions for publication in IMAGE. Meanwhile we still lookforward to receiving solutions to unsolved problems in the past issues of IMAGE: Problems 36-3 [IMAGE 36 (Spring 2006), p. 36] and 39-4[IMAGE 39 (Fall 2007), p. 32]. Submissions: Please submit proposed problems and solutions in macro-free LATEX along with the PDF file by e-mailto IMAGE Problem Corner editor Fuzhen Zhang ([email protected]). The working team of the Problem Corner consists of Dennis S. Bernstein, NirCohen, Shaun Fallat, Dennis Merino, Edward Poon, Peter Semrl, Wasin So, Nung-Sing Sze, and Xingzhi Zhan.

NEW PROBLEMS:

Problem 43-1: Characterization of EP MatricesProposed by Oskar Maria Baksalary, Adam Mickiewicz University, Poznan, Poland, [email protected] Gotz Trenkler, Technische Universitat Dortmund, Dortmund, Germany, [email protected]

Let A be an n× n complex matrix. Show that A is EP (i.e., AA† = A†A, where A† is the Moore-Penrose inverse of A) if and onlyif the column space of A2 coincides with the column space of the conjugate transpose of A.

Problem 43-2: Square Roots of (Skew-)Involutory MatricesProposed by Richard William Farebrother, Bayston Hill, Shrewsbury, England, [email protected]

Identify a family of square matrices with integral elements (few of them zeros) whose squares are nonzero multiples of involutoryor skew-involutory matrices. [Recollect that an n× n matrix A is said to be involutory if it satisfies A2 = In and skew-involutory ifit satisfies A2 = −In.]

Problem 43-3: Reconstructing a Positive Semidefinite MatrixProposed by Bryan T. Kelly, NYU Stern School of Business, New York, USAand Chi-Kwong Li, College of William and Mary, Williamsburg, USA, [email protected]

SupposeA = (Ars)1≤r,s≤k is a block complex Hermitian (or real symmetric) positive semidefinite matrix such thatArr is nr-by-nr.Show that A is positive semidefinite if A is obtained from A as follows:

(a) For r 6= s, replace each entry of Ars by the average of its entries, and(b) Replace each diagonal entry of each diagonal block Arr by the average of its diagonal entries, and replace each off-diagonal

entry of Arr by the average of its off-diagonal entries.

Problem 43-4: A Trace Inequality for Positive Semidefinite MatricesProposed by Minghua Lin, University of Regina, Saskatchewan, Canada, [email protected]

Let A be positive semidefinite and B be positive definite. Show that tr(Ap+1B−p) ≥ (trA)p+1(trB)−p for any positive integer p.

Problem 43-5: The Matrix Equation Y TY = YProposed by Heinz Neudecker, University of Amsterdam, Amsterdam, The Netherlands, [email protected]

Let Y and T be n × n real matrices and 1 be the n-column vector of ones. Assume that Y is positive semidefinite, of rank n − 1,Y 1 = 0, Y TY = Y , and that T is symmetric and has full rank. Find a relationship between 1′T−11 and detT that shows they havethe same sign. As usual, detT is the determinant of the matrix T .

Problem 43-6: Matrix SimilarityProposed by Xingzhi Zhan, East China Normal University, Shanghai, China, [email protected]

Let A and B be complex matrices of the same order. If A and B are similar and *-congruent and have the same value of Frobeniusnorm, then does it follow that A and B are unitarily similar?

Problem 43-7: Three Positive Definite MatricesProposed by Fuzhen Zhang, Nova Southeastern University, Fort Lauderdale, USA, [email protected]

It is known that any two positive semidefinite matrices of the same size are simultaneously ∗-congruent to diagonal matrices. Canthis be generalized to three positive semidefinite matrices? That is, if A, B, and C are positive semidefinite matrices, does therealways exist an invertible matrix P such that P ∗AP , P ∗BP , and P ∗CP are all diagonal?

Solutions to Problems 41-4 and 42-1 through 42-6 are on page 35.

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